Quick recall · 254 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
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Which of the following shows the correct conversion of 2.4 to a mixed fraction?
A) \( 2\frac{1}{5} \)
B) \( 2\frac{2}{5} \)
C) \( 2\frac{3}{5} \)
D) \( 2\frac{4}{5} \)
B · \( 2\frac{2}{5} \)
For decimal 2.4, consider 0.4 part: \( \frac{0.4}{1} \).One decimal place, multiply numerator and denominator by 10: \( \frac{4}{10} = \frac{2}{5} \).Thus, \( 2\frac{2}{5} \). Option B matches.
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Which number is divisible by 2?
C · 46
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Which of the following numbers are divisible by 5? 10000, 2255, 65, 80, 925
D · Only 10000, 2255, 65, 80, and 925
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The number 21A35B4 is divisible by 3, where A and B are non-zero digits. What is the maximum possible value for A + B?
B · 15
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A two-digit number 'X' is divisible by 2, 3, and 5. Which of the following is definitely wrong about 'X'?
D · X is an odd number
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Simplify: 3 + 4 × 2
A · 11
According to BODMAS rule, multiplication is performed before addition. First calculate 4 × 2 = 8, then 3 + 8 = 11. Therefore, the correct answer is option **A** (11).[1]
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If a : b = 5 : 3, what percentage of 3a is (3a + 4b)?
A · 50%
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A and B together have Rs. 1210. If \( \frac{3}{5} \) of A's amount is equal to \( \frac{4}{5} \) of B's amount, how much amount does B have?
A · Rs. 2000
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What is the cube root of 2197?
B · B. 13
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The cube root of 0.000216 is:
B · B. 0.06
Rewrite 0.000216 as \(216 \times 10^{-6} = (6^3) \times (10^{-2})^3 = (6 \times 10^{-2})^3 = 0.06^3\). Thus, \(\sqrt[3]{0.000216} = 0.06\). Option B is correct.
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Which of the following is a natural number?
C · 7
Natural numbers are positive integers starting from 1, so 7 is a natural number.
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What is the smallest natural number?
A · 1
By definition, natural numbers start from 1 upwards.
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Which of the following is NOT a natural number?
C · 0
Zero is not considered a natural number in the standard definition.
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Find the sum of the first 10 natural numbers.
A · 55
Sum of first n natural numbers is \( \frac{n(n+1)}{2} \). For n=10, sum = \( \frac{10 \times 11}{2} = 55 \).
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Which of the following statements is TRUE about natural numbers?
C · Natural numbers are whole numbers greater than zero
Natural numbers are whole numbers starting from 1 upwards, excluding zero and negatives.
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Which of the following integers is NOT a whole number?
B · -3
Whole numbers are non-negative integers including zero, so -3 is not a whole number.
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What is the result of \( -7 + 12 \)?
A · 5
Adding -7 and 12 equals 5.
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If \( x = -3 \) and \( y = 5 \), what is \( x \times y \)?
A · -15
Multiplying a negative and positive integer results in a negative product: \( -3 \times 5 = -15 \).
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Which integer lies exactly halfway between -4 and 6 on the number line?
A · 1
Midpoint = \( \frac{-4 + 6}{2} = \frac{2}{2} = 1 \).
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What is the value of \( |-8| + |5| \)?
B · 13
Absolute values: \( |-8| = 8 \), \( |5| = 5 \), sum = 13.
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Which of the following is NOT an integer?
C · 3.5
3.5 is a decimal number, not an integer.
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Simplify \( \frac{18}{24} \) to its lowest terms.
A · \( \frac{3}{4} \)
GCD of 18 and 24 is 6, so \( \frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \).
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Which of the following fractions is equivalent to \( \frac{5}{10} \)?
A · \( \frac{1}{2} \)
\( \frac{5}{10} = \frac{1}{2} \) after dividing numerator and denominator by 5.
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Calculate \( \frac{3}{4} + \frac{2}{5} \).
A · \( \frac{23}{20} \)
Find common denominator 20: \( \frac{3}{4} = \frac{15}{20}, \frac{2}{5} = \frac{8}{20} \). Sum = \( \frac{15+8}{20} = \frac{23}{20} \).
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Which fraction is greater than \( \frac{3}{7} \)?
C · \( \frac{4}{7} \)
\( \frac{4}{7} > \frac{3}{7} \) because numerator is larger with same denominator.
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What is the product of \( \frac{2}{3} \) and \( \frac{9}{4} \)?
A · \( \frac{3}{2} \)
Multiply numerators and denominators: \( \frac{2 \times 9}{3 \times 4} = \frac{18}{12} = \frac{3}{2} \) after simplification.
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Simplify \( \frac{45}{60} \) and express as a decimal.
A · 0.75
Simplify fraction: \( \frac{45}{60} = \frac{3}{4} = 0.75 \).
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Convert the decimal 0.625 to a fraction in simplest form.
A · \( \frac{5}{8} \)
0.625 = \( \frac{625}{1000} = \frac{5}{8} \) after simplification.
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Which decimal is equivalent to \( \frac{7}{10} \)?
A · 0.7
\( \frac{7}{10} = 0.7 \).
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What is the value of \( 3.5 + 2.75 \)?
A · 6.25
Adding decimals: 3.5 + 2.75 = 6.25.
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Which decimal is greater than 0.45?
B · 0.54
0.54 > 0.45, others are less.
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Express 0.375 as a fraction in simplest form.
A · \( \frac{3}{8} \)
0.375 = \( \frac{375}{1000} = \frac{3}{8} \) after simplification.
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Calculate \( 5.2 \times 3.1 \).
A · 16.12
Multiplying decimals: 5.2 \times 3.1 = 16.12 (incorrect), correct is 16.12. However, 5.2 \times 3.1 = 16.12.
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Which of the following decimals is a terminating decimal?
B · 0.25
0.25 is terminating; others are repeating decimals.
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Which property states that \( a + b = b + a \)?
B · Commutative Property
The Commutative Property states that the order of addition or multiplication does not change the result.
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What is the result of \( (2 + 3) + 4 \) using associative property?
A · 9
Associative property: \( (2 + 3) + 4 = 2 + (3 + 4) = 9 \).
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Simplify \( 5 \times (2 + 3) \) using distributive property.
B · 10 + 15
Distributive property: \( 5 \times (2 + 3) = 5 \times 2 + 5 \times 3 = 10 + 15 \).
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Which of the following is the identity element for multiplication?
B · 1
Multiplying any number by 1 leaves it unchanged, so 1 is the identity element.
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If \( a = 4 \) and \( b = 0 \), what is \( a + b \)?
A · 4
Adding zero to any number leaves it unchanged (additive identity).
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Convert \( \frac{3}{8} \) to decimal form.
A · 0.375
\( \frac{3}{8} = 0.375 \) in decimal.
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Express 0.2 as a fraction in simplest form.
A · \( \frac{1}{5} \)
0.2 = \( \frac{2}{10} = \frac{1}{5} \) after simplification.
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Convert the repeating decimal \( 0.\overline{6} \) to a fraction.
A · \( \frac{2}{3} \)
The repeating decimal 0.666... equals \( \frac{2}{3} \).
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Express \( \frac{7}{20} \) as a decimal.
A · 0.35
\( \frac{7}{20} = 0.35 \) as a decimal.
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Convert the fraction \( \frac{11}{30} \) to a decimal (rounded to 3 decimal places).
A · 0.367
\( \frac{11}{30} = 0.3666... \) rounded to 0.367.
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Simplify the expression \( \frac{2}{3} \times \frac{9}{4} \) and write the answer in simplest form.
A · \( \frac{3}{2} \)
Multiply: \( \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2} \) after simplification.
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Simplify \( \frac{36}{48} \) using the highest common factor.
A · \( \frac{3}{4} \)
GCF of 36 and 48 is 12, so \( \frac{36}{48} = \frac{3}{4} \).
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Which of the following fractions is in simplest form?
B · \( \frac{7}{13} \)
\( \frac{7}{13} \) is already in simplest form as 7 and 13 have no common factors other than 1.
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Simplify the expression \( \frac{5}{6} - \frac{1}{3} \).
A · \( \frac{1}{2} \)
Convert to common denominator 6: \( \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \).
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Which of the following is a natural number?
C · 5
Natural numbers are positive integers starting from 1, so 5 is a natural number.
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What is the sum of the first 10 natural numbers?
A · 55
Sum of first n natural numbers is \( \frac{n(n+1)}{2} \). For n=10, sum = \( \frac{10 \times 11}{2} = 55 \).
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Which of the following statements is true about natural numbers?
B · They are all positive integers excluding zero
Natural numbers are positive integers starting from 1, excluding zero and negative numbers.
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If \( n \) is a natural number, which of the following is always even?
B · \( 2n \)
Multiplying any natural number by 2 results in an even number.
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Find the 7th natural number after 15.
B · 22
The 7th natural number after 15 is \( 15 + 7 = 22 \). However, since the question asks for the 7th natural number after 15, it means counting 7 numbers after 15, so the answer is 22.
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Which of the following is NOT a property of natural numbers?
B · Closure under subtraction
Natural numbers are not closed under subtraction because subtracting a larger number from a smaller one results in a negative number, which is not a natural number.
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Which integer lies exactly midway between \(-7\) and \(5\)?
A · -1
Midpoint = \( \frac{-7 + 5}{2} = \frac{-2}{2} = -1 \).
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Which of the following is NOT an integer?
C · 4.5
Integers are whole numbers including negatives, zero, and positives. 4.5 is a decimal and not an integer.
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Evaluate \( (-3) \times (-4) + 5 \).
B · 17
Multiplying two negatives gives a positive: \( (-3) \times (-4) = 12 \). Adding 5 gives \( 12 + 5 = 17 \).
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Which integer is the additive inverse of \(-15\)?
A · 15
The additive inverse of a number is the number which when added to it gives zero. So, additive inverse of \(-15\) is 15.
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Simplify: \( |-8| + (-3) \).
A · 5
Absolute value of \(-8\) is 8. So, \( 8 + (-3) = 5 \).
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If \( x = -4 \), what is the value of \( 3x^2 - 2x + 1 \)?
A · 57
Calculate \( 3(-4)^2 - 2(-4) + 1 = 3(16) + 8 + 1 = 48 + 8 + 1 = 57 \).
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Which of the following fractions is equivalent to \( \frac{3}{4} \)?
A · \( \frac{6}{8} \)
\( \frac{6}{8} = \frac{3 \times 2}{4 \times 2} = \frac{3}{4} \).
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What is \( \frac{5}{6} + \frac{1}{3} \)?
A · \( \frac{7}{6} \)
Convert \( \frac{1}{3} \) to \( \frac{2}{6} \). Sum = \( \frac{5}{6} + \frac{2}{6} = \frac{7}{6} \).
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Simplify \( \frac{12}{18} \) to its lowest terms.
A · \( \frac{2}{3} \)
Divide numerator and denominator by 6: \( \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \).
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Which of the following is the product of \( \frac{3}{5} \) and \( \frac{10}{9} \)?
A · \( \frac{2}{3} \)
Multiply numerators and denominators: \( \frac{3 \times 10}{5 \times 9} = \frac{30}{45} = \frac{2}{3} \).
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If \( \frac{a}{8} = \frac{3}{4} \), what is the value of \( a \)?
A · 6
Cross multiply: \( 4a = 3 \times 8 = 24 \) so \( a = 6 \).
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Which fraction is greater than \( \frac{2}{5} \)?
B · \( \frac{1}{2} \)
\( \frac{1}{2} = 0.5 \) which is greater than \( \frac{2}{5} = 0.4 \).
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Simplify \( \frac{7}{8} - \frac{3}{4} \).
A · \( \frac{1}{8} \)
Convert \( \frac{3}{4} = \frac{6}{8} \). Subtract: \( \frac{7}{8} - \frac{6}{8} = \frac{1}{8} \).
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Convert the decimal 0.375 to a fraction in simplest form.
A · \( \frac{3}{8} \)
0.375 = \( \frac{375}{1000} = \frac{3}{8} \) after simplification.
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What is the value of \( 5.6 + 3.45 \)?
A · 9.05
Add the decimals: 5.6 + 3.45 = 9.05.
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Which decimal is equivalent to \( \frac{7}{20} \)?
A · 0.35
\( \frac{7}{20} = 7 \div 20 = 0.35 \).
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Round 4.6789 to two decimal places.
B · 4.68
The third decimal digit is 8 (greater than 5), so round up the second decimal place from 7 to 8.
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Subtract \( 2.75 \) from \( 5.6 \).
A · 2.85
5.6 - 2.75 = 2.85.
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Express \( 0.2\overline{3} \) (0.2333...) as a fraction in simplest form.
A · \( \frac{7}{30} \)
Let \( x = 0.2\overline{3} \). Multiply by 10: \( 10x = 2.3\overline{3} \). Subtract \( x \): \( 9x = 2.1 \) so \( x = \frac{21}{90} = \frac{7}{30} \).
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Convert \( \frac{11}{25} \) to a decimal.
A · 0.44
\( \frac{11}{25} = 11 \div 25 = 0.44 \).
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Which fraction corresponds to the decimal 0.125?
A · \( \frac{1}{8} \)
0.125 = \( \frac{125}{1000} = \frac{1}{8} \) after simplification.
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Convert the repeating decimal \( 0.\overline{6} \) to a fraction.
A · \( \frac{2}{3} \)
Repeating decimal \( 0.\overline{6} = \frac{2}{3} \).
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Express \( \frac{5}{8} \) as a decimal.
A · 0.625
\( \frac{5}{8} = 5 \div 8 = 0.625 \).
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Which decimal is equal to \( \frac{9}{20} \)?
A · 0.45
\( \frac{9}{20} = 9 \div 20 = 0.45 \).
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Simplify the expression \( 3 + 5 \times 2 - 4 \div 2 \).
B · 11
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Evaluate \( (8 - 3)^2 \div 5 + 6 \).
A · 11
Calculate inside parentheses: \( 8 - 3 = 5 \). Then square: \( 5^2 = 25 \). Divide: \( 25 \div 5 = 5 \). Add 6: \( 5 + 6 = 11 \). So correct answer is 11.
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Simplify \( \frac{3}{4} + \frac{2}{3} \times \frac{6}{5} \).
D · \( \frac{31}{20} \)
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Calculate \( 12 - 3 \times (2 + 4) \div 3 \).
A · 6
Calculate inside parentheses: \( 2 + 4 = 6 \). Multiply: \( 3 \times 6 = 18 \). Divide: \( 18 \div 3 = 6 \). Subtract: \( 12 - 6 = 6 \).
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Simplify \( \frac{5}{6} \div \frac{10}{9} + 2 \).
C · \( \frac{8}{3} \)
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Evaluate \( 4.5 \times (2 + 3.5) - 6 \).
A · 23.25
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Assertion (A): If \(x, y, z\) are integers such that \(x + y + z = 0\) and \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0\), then \(x^2 + y^2 + z^2 = 0\).
Reason (R): The given conditions imply that \(xy + yz + zx = 0\).
D · A is false but R is true
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If \(x, y, z\) are natural numbers satisfying \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1\) and \(x < y < z\), which of the following triples \((x,y,z)\) is a solution?
A · (2, 3, 6)
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Consider the decimal \(0.abcabcabc...\) where \(a,b,c\) are digits with \(a eq 0\). If \(\frac{p}{q} = 0.abcabcabc...\) in simplest form, and \(p+q= 143\), what is the value of \(a+b+c\)?
B · 12
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If \(m, n\) are integers such that \(\frac{m}{n} = 0.\overline{123456}\) and \(m + n = 1234\), what is the value of \(m - n\)?
B · 246
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Which of the following statements is TRUE about the number \(\frac{7}{11} + 0.\overline{63}\)?
D · It is a rational number with denominator 33 in simplest form
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If \(x\) is a natural number such that \(\frac{1}{x} + \frac{1}{x+1} = \frac{3}{5}\), what is the value of \(x^2 + (x+1)^2\)?
C · 109
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Which of the following fractions is closest to the decimal \(0.142857\) but is NOT equal to \(\frac{1}{7}\)?
C · \(\frac{5}{35}\)
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If \(a, b\) are integers such that \(\frac{a}{b} = 0.\overline{09}\), which of the following is TRUE?
A · \(a = 1, b = 11\)
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Assertion (A): The decimal \(0.1\overline{6}\) is equal to \(\frac{1}{6}\).
Reason (R): \(0.1\overline{6}\) can be expressed as \(\frac{1}{10} + \frac{6}{90}\).
B · Both A and R are true but R is not the correct explanation of A
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Find the value of \(k\) if \(\frac{3k + 1}{2k - 1} = 1.25\) and \(k\) is an integer.
B · 5
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If \(x\) is a natural number such that \(\frac{1}{x} + \frac{1}{x+2} = \frac{5}{12}\), find \(x\).
B · 4
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Which of the following fractions has a terminating decimal expansion?
A · \(\frac{7}{20}\)
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If \(\frac{p}{q} = 0.\overline{abc}\) where \(abc\) is a three-digit number and \(p+q= 143\), which of the following could be \(abc\)?
C · 333
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Find the sum of all natural numbers \(n\) such that \(\frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} = 1\).
C · 9
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What is the Least Common Multiple (LCM) of 6 and 8?
B · 24
The multiples of 6 are 6, 12, 18, 24, ... and the multiples of 8 are 8, 16, 24, ... The smallest common multiple is 24.
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Which of the following is NOT a property of LCM?
D · LCM of two numbers divides their product
The LCM of two numbers does not necessarily divide their product; rather, the product of the two numbers equals the product of their LCM and HCF.
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If the LCM of two numbers is 60 and one of the numbers is 12, which of the following could be the other number?
C · 15
LCM(12, x) = 60. Since 60 is divisible by 12, the other number must be a divisor of 60 such that LCM is 60. 15 fits because LCM(12,15) = 60.
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What is the Highest Common Factor (HCF) of 18 and 24?
A · 6
The factors of 18 are 1, 2, 3, 6, 9, 18 and factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The greatest common factor is 6.
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Which of the following is TRUE about the HCF of two numbers?
B · HCF of two prime numbers is always 1
Two prime numbers have no common factors other than 1, so their HCF is always 1.
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If the HCF of two numbers is 4 and one of the numbers is 12, which of the following could be the other number?
A · 16
HCF(12, x) = 4. The other number must be divisible by 4 but not share a higher common factor with 12. 16 fits because HCF(12,16) = 4.
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Find the LCM of 18 and 24 using prime factorization.
A · 72
Prime factors: 18 = 2 \times 3^2, 24 = 2^3 \times 3. LCM takes highest powers: 2^3 \times 3^2 = 8 \times 9 = 72.
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Using the division method, find the HCF of 48 and 60.
B · 12
Divide both by common prime factors: 48 and 60 are divisible by 2, then 2, then 3. So HCF = 2 \times 2 \times 3 = 12.
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If the HCF of two numbers is 5 and their LCM is 180, and one number is 45, what is the other number?
A · 20
Using the relation: Product = HCF \times LCM = 5 \times 180 = 900. Given one number is 45, other number = 900 / 45 = 20, but 20 and 45 have HCF 5? 20 and 45 HCF is 5. So correct answer is 20.
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Which equation correctly represents the relationship between LCM and HCF of two numbers \(a\) and \(b\)?
B · \( \text{LCM}(a,b) \times \text{HCF}(a,b) = a \times b \)
The product of LCM and HCF of two numbers equals the product of the numbers themselves.
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If the HCF of two numbers is 7 and their product is 1470, what is their LCM?
A · 210
Using \( \text{LCM} = \frac{a \times b}{\text{HCF}} = \frac{1470}{7} = 210 \).
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Two numbers have an HCF of 6 and an LCM of 180. If one number is 54, what is the other number?
C · 30
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Two gears with 12 and 18 teeth start together. After how many rotations of the smaller gear will they align again at the starting point?
D · 36
The gears align after LCM of number of teeth rotations. LCM(12,18) = 36.
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Three traffic lights flash at intervals of 12, 15, and 20 seconds respectively. If they all flash together at 8:00 AM, when will they next flash together?
A · 8:01 AM
Find LCM of 12, 15, and 20: prime factors are 2^2, 3, 5. LCM = 60 seconds = 1 minute. So next flash together at 8:01 AM.
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A school bus leaves the school every 15 minutes, and a train passes the school every 20 minutes. If both leave at 9:00 AM, after how many minutes will they leave together again?
B · 60
Find LCM of 15 and 20. 15 = 3 \times 5, 20 = 2^2 \times 5, so LCM = 2^2 \times 3 \times 5 = 60 minutes.
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What is the least common multiple (LCM) of 6 and 8?
B · 24
The multiples of 6 are 6, 12, 18, 24, ... and the multiples of 8 are 8, 16, 24, ... The smallest common multiple is 24.
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Which of the following is NOT a property of the LCM of two numbers?
C · LCM is always equal to the product of the two numbers
LCM is not always equal to the product of two numbers; it is equal to the product divided by their HCF.
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If the LCM of two numbers is 180 and one of the numbers is 36, which of the following could be the other number?
B · 30
LCM(36, x) = 180. Since 36 = 2^2 * 3^2 and 180 = 2^2 * 3^2 * 5, the other number must include the factor 5 but not reduce the LCM. 30 = 2 * 3 * 5 fits.
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What is the highest common factor (HCF) of 48 and 60?
B · 12
Prime factors of 48 are 2^4 * 3, and of 60 are 2^2 * 3 * 5. The common factors are 2^2 * 3 = 12.
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Which of the following is true about the HCF of two numbers?
A · HCF is always less than or equal to both numbers
HCF is the greatest number that divides both numbers, so it cannot be greater than the smaller number.
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If the HCF of two numbers is 7 and one of the numbers is 35, which of the following could be the other number?
B · 21
HCF(35, x) = 7. 35 = 7 * 5, so the other number must have 7 but not 5. 21 = 7 * 3 fits.
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Using the Euclidean algorithm, what is the HCF of 56 and 98?
B · 14
98 ÷ 56 = 1 remainder 4256 ÷ 42 = 1 remainder 1442 ÷ 14 = 3 remainder 0So, HCF is 14.
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Find the LCM of 15 and 20 using prime factorization.
A · 60
15 = 3 * 520 = 2^2 * 5LCM = 2^2 * 3 * 5 = 60
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If the product of two numbers is 360 and their HCF is 6, what is their LCM?
C · 60
Product = HCF × LCMSo, LCM = Product ÷ HCF = 360 ÷ 6 = 60
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Which equation correctly shows the relationship between two numbers \(a\) and \(b\), their LCM, and HCF?
C · \( a \times b = \text{LCM}(a,b) \times \text{HCF}(a,b) \)
The product of two numbers equals the product of their LCM and HCF.
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Two machines operate on cycles of 12 and 18 minutes respectively. After how many minutes will they both operate together again?
A · 36
LCM of 12 and 18 is 36, so they will operate together after 36 minutes.
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A gardener wants to plant flowers in rows such that each row has the same number of flowers and uses all 48 red and 60 yellow flowers. What is the greatest number of flowers in each row?
B · 12
HCF of 48 and 60 is 12, so the greatest number of flowers per row is 12.
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Simplify the expression \( \frac{24}{36} + \frac{15}{45} \) using the LCM of denominators.
D · \( \frac{11}{12} \)
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Simplify the expression \( \frac{2}{3} \times \frac{9}{4} \) by first finding the HCF of numerator and denominator terms.
A · \( \frac{3}{2} \)
Multiply numerators and denominators:\( \frac{2 \times 9}{3 \times 4} = \frac{18}{12} \).HCF of 18 and 12 is 6.Simplify: \( \frac{18 \div 6}{12 \div 6} = \frac{3}{2} \).
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Using the division method, find the HCF of 84 and 126.
C · 42
Divide both numbers by common prime factors:84 ÷ 2 = 42, 126 ÷ 2 = 6342 ÷ 3 = 14, 63 ÷ 3 = 2114 ÷ 7 = 2, 21 ÷ 7 = 3HCF = 2 × 3 × 7 = 42
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A bus arrives at a stop every 12 minutes and a train every 18 minutes. If they both arrive together now, after how many minutes will they next arrive together?
B · 36
The time after which both arrive together is the LCM of 12 and 18, which is 36 minutes.
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If two numbers x and y satisfy the conditions:
(i) HCF(x, y) = d,
(ii) LCM(x, y) = 4620,
(iii) x + y = 462 + d,
where d divides 4620, find the value of d.
B · 21
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Match the following pairs where each pair consists of (Number, Number of divisors of its LCM with 84):
(A) 36
(B) 60
(C) 105
(D) 140
Options:
1) 24
2) 18
3) 20
4) 30
Find the correct matching.
A · (A-2, B-4, C-3, D-1)
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If three positive integers x, y, z satisfy:
(i) HCF(x,y) = 12,
(ii) HCF(y,z) = 18,
(iii) HCF(x,z) = 6,
(iv) LCM(x,y,z) = 2^4 × 3^3 × 5,
then what is the minimum possible value of x + y + z?
B · 2^3 × 3^3 × 5
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Two numbers A and B satisfy the following:
(i) LCM(A,B) = 2^5 × 3^3 × 7,
(ii) HCF(A,B) = 2^2 × 3 × 7^2,
(iii) A + B = 2^6 × 3^2 × 7^3.
Find the value of A × B.
A · 2^7 × 3^4 × 7^4
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If positive integers m and n satisfy:
(i) HCF(m,n) = 1,
(ii) LCM(m,n) = 4620,
(iii) m^2 + n^2 = 2 × LCM(m,n),
then find the value of m + n.
D · 462
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For positive integers x and y, the following hold:
(i) HCF(x,y) = 1,
(ii) LCM(x,y) = 2^3 × 3^2 × 5 × 7,
(iii) x + y = 2^4 × 3 × 5 × 7.
Find the value of |x - y|.
A · 2^3 × 3 × 5 × 7
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If three positive integers a, b, c satisfy:
(i) HCF(a,b) = 15,
(ii) HCF(b,c) = 21,
(iii) HCF(a,c) = 35,
(iv) LCM(a,b,c) = 2^3 × 3^2 × 5 × 7^2,
then find the minimum possible value of a × b × c.
A · 2^3 × 3^2 × 5^3 × 7^3
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If positive integers p and q satisfy:
(i) HCF(p,q) = 1,
(ii) LCM(p,q) = 2^4 × 3^3 × 5^2,
(iii) p^2 + q^2 = 2 × LCM(p,q),
then find the value of p × q.
A · 2^4 × 3^3 × 5^2
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If positive integers a and b satisfy:
(i) HCF(a,b) = 1,
(ii) LCM(a,b) = 2^3 × 3^2 × 5,
(iii) a + b = 2^4 × 3 × 5,
then find the value of (a - b)^2.
A · 2^6 × 3^2 × 5^2 - 4 × 2^3 × 3^2 × 5
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Match the following:
Set 1: Numbers
(A) 84
(B) 90
(C) 105
(D) 126
Set 2: Number of divisors of their HCF with 210
Options:
1) 6
2) 8
3) 4
4) 12
Find the correct matching.
A · (A-2, B-1, C-3, D-4)
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If positive integers x, y satisfy:
(i) HCF(x,y) = 6,
(ii) LCM(x,y) = 180,
(iii) x + y = 54,
then find the value of x - y.
B · 12
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If positive integers a, b satisfy:
(i) HCF(a,b) = 4,
(ii) LCM(a,b) = 180,
(iii) a + b = 52,
then find the value of (a^2 + b^2).
A · 1360
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Which of the following numbers is divisible by 3?
B · 135
A number is divisible by 3 if the sum of its digits is divisible by 3. For 135, the sum is 1 + 3 + 5 = 9, which is divisible by 3.
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Which of the following numbers is divisible by 5?
A · 230
A number is divisible by 5 if it ends with 0 or 5. 230 ends with 0, so it is divisible by 5.
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Is the number 2730 divisible by both 2 and 9?
A · Yes, divisible by both
2730 ends with 0, so divisible by 2. Sum of digits = 2 + 7 + 3 + 0 = 12, which is divisible by 9. Hence divisible by both.
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Which of the following numbers is divisible by 4?
A · 1232
A number is divisible by 4 if its last two digits form a number divisible by 4. 32 is divisible by 4, so 1232 is divisible by 4.
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Which of the following numbers is divisible by 6?
A · 234
A number is divisible by 6 if it is divisible by both 2 and 3. 234 ends with 4 (even) and sum of digits is 2+3+4=9 (divisible by 3).
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Is the number 123456 divisible by 8?
A · Yes
A number is divisible by 8 if its last three digits form a number divisible by 8. Last three digits are 456; 456 ÷ 8 = 57 with remainder 0, so divisible by 8. Hence answer should be Yes.
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Find the remainder when 12345 is divided by 11.
B · 1
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A number is divisible by 2, 3, and 5. Which of the following could be the smallest such number?
C · 30
The number must be divisible by the LCM of 2, 3, and 5. LCM(2,3,5) = 30.
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If a number is divisible by both 4 and 6, which of the following must it be divisible by?
A · 12
The number must be divisible by the LCM of 4 and 6, which is 12.
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A 5-digit number \( \overline{abcde} \) is divisible by 11. If \( a = 7, b = 4, c = 8, d = 3 \), what is the value of \( e \)?
C · 6
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Which of the following numbers is divisible by 2?
B · 24681
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Which of the following numbers is divisible by 2?
A · 12344
A number is divisible by 2 if its last digit is even. 12344 ends with 4, which is even, so it is divisible by 2.
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Which of the following numbers is divisible by 5?
A · 12340
A number is divisible by 5 if it ends with 0 or 5. 12340 ends with 0, so it is divisible by 5.
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Is the number 123456 divisible by 3?
A · Yes, because the sum of digits is divisible by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. Sum of digits = 1+2+3+4+5+6 = 21, which is divisible by 3.
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Which of the following numbers is divisible by 4?
A · 1232
A number is divisible by 4 if the last two digits form a number divisible by 4. Last two digits of 1232 are 32, which is divisible by 4.
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Which of the following numbers is divisible by 11?
A · 2728
For divisibility by 11, subtract sum of digits in odd positions and even positions. For 2728: (2+2) - (7+8) = 4 - 15 = -11, which is divisible by 11.
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Is the number 123456 divisible by 6?
A · Yes, because it is divisible by both 2 and 3
A number is divisible by 6 if it is divisible by both 2 and 3. 123456 ends with 6 (even) and sum of digits is 21 (divisible by 3), so divisible by 6.
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Which of the following numbers is divisible by 8?
C · 123440
A number is divisible by 8 if the last three digits form a number divisible by 8. Last three digits of 123440 are 440, and 440 ÷ 8 = 55, so divisible by 8.
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If a number is divisible by both 3 and 5, which of the following must it be divisible by?
A · 15
If a number is divisible by both 3 and 5, it must be divisible by their least common multiple, which is 15.
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Which of the following numbers is divisible by 9?
D · 123444
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Which of the following numbers is divisible by 9?
D · 123444
Sum of digits of 123444 = 1+2+3+4+4+4 = 18, which is divisible by 9, so the number is divisible by 9.
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Which of the following numbers is divisible by 11?
A · 121
For divisibility by 11, difference between sum of digits in odd and even positions should be 0 or multiple of 11. For 121: (1+1) - (2) = 2 - 2 = 0, divisible by 11.
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Which of the following numbers is divisible by both 2 and 3?
B · 144
A number divisible by both 2 and 3 is divisible by 6. 144 ends with 4 (even) and sum of digits is 1+4+4=9, divisible by 3, so divisible by 6.
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If the sum of digits of a number is 27, which of the following can be true?
B · The number is divisible by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. Since 27 is divisible by 9, the number is divisible by 9.
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Find the smallest 4-digit number divisible by both 4 and 6.
A · 1008
Number divisible by both 4 and 6 must be divisible by LCM(4,6) = 12. The smallest 4-digit number divisible by 12 is 1008.
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Which of the following numbers is divisible by 18?
D · 540
A number is divisible by 18 if it is divisible by both 9 and 2. 540 ends with 0 (even) and sum of digits is 5+4+0=9, divisible by 9, so divisible by 18.
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If the difference between the sum of digits in odd and even positions of a number is 33, which of the following is true?
A · The number is divisible by 11
A number is divisible by 11 if the difference between the sum of digits in odd and even positions is 0 or a multiple of 11. 33 is a multiple of 11, so the number is divisible by 11.
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Which of the following numbers is divisible by 4 based on digit manipulation?
A · Number ending with 12
A number is divisible by 4 if the last two digits form a number divisible by 4. 12 is divisible by 4, so numbers ending with 12 are divisible by 4.
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Which of the following correctly represents the order of operations in the BODMAS rule?
A · Brackets, Orders, Division, Multiplication, Addition, Subtraction
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In the expression \( 8 + 2 \times (5 - 3)^2 \), which operation should be performed first according to BODMAS?
B · Subtraction inside the bracket
According to BODMAS, operations inside brackets are performed first. So, subtraction inside the bracket \( (5 - 3) \) is done before exponentiation and multiplication.
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Simplify the expression \( 6 + 4 \times 3 - 8 \div 2 \) using BODMAS.
B · 14
First, multiplication and division from left to right: \(4 \times 3 = 12\), \(8 \div 2 = 4\). Then addition and subtraction: \(6 + 12 - 4 = 14\). So the correct answer is 14.
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Evaluate \( (12 - 4) \div 2 + 3^2 \) using BODMAS.
A · 13
Calculate inside brackets first: \(12 - 4 = 8\). Then division: \(8 \div 2 = 4\). Then exponentiation: \(3^2 = 9\). Finally, addition: \(4 + 9 = 13\). So the correct answer is 13.
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Simplify the expression \( 18 \div (3 \times (2 + 1)) \).
A · 2
First solve the innermost bracket: \(2 + 1 = 3\). Then multiply: \(3 \times 3 = 9\). Finally, divide: \(18 \div 9 = 2\).
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Evaluate \( \left[ 5 + (3^2 - 4) \right] \times 2 \).
B · 26
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Simplify \( \left( 2 + \left[ 3 \times (4 + 1) \right] \right) \div 5 \).
A · 4
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Calculate \( 2^3 \times 3^2 \) using BODMAS.
A · 72
Calculate exponents first: \(2^3 = 8\), \(3^2 = 9\). Then multiply: \(8 \times 9 = 72\).
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Simplify \( \sqrt{16} + 2^3 \).
A · 12
Calculate square root: \(\sqrt{16} = 4\). Calculate exponent: \(2^3 = 8\). Add: \(4 + 8 = 12\). Since 12 is not option B, re-check options. Option A is 12, so correct answer is A.
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Evaluate \( \left( 3 + 5 \right)^2 \div \sqrt{9} \).
A · 21
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Simplify \( \frac{1}{2} + \frac{3}{4} \times 2 \).
C · 2
Multiply first: \(\frac{3}{4} \times 2 = \frac{6}{4} = \frac{3}{2}\). Then add: \(\frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2\). Since 2 is option C, correct answer is C.
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Calculate \( 0.5 + 1.2 \times (0.8 + 0.2) \) using BODMAS.
A · 1.7
Calculate inside bracket: \(0.8 + 0.2 = 1.0\). Multiply: \(1.2 \times 1.0 = 1.2\). Add: \(0.5 + 1.2 = 1.7\). Since 1.7 is option A, correct answer is A.
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Which of the following correctly represents the order of operations in the BODMAS rule?
A · Brackets, Orders, Division, Multiplication, Addition, Subtraction
BODMAS stands for Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction, and operations are performed in this order.
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In the expression \( 8 + 2 \times 5 - 3 \), what is the correct result after applying BODMAS?
B · 15
According to BODMAS, multiplication is done before addition and subtraction: \(2 \times 5 = 10\), then \(8 + 10 - 3 = 15\). So the correct answer is 15.
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Evaluate \( 12 - 4 \times (3 + 2)^2 \div 2 \) using BODMAS.
B · -38
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Simplify \( 7 + 3 \times 2^3 - 4 \div 2 \) using BODMAS.
C · 19
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Simplify \( \left[ 5 + \{ 3 \times (2 + 4) \} \right] - 7 \) using BODMAS.
A · 16
Innermost bracket: \(2 + 4 = 6\). Then multiplication: \(3 \times 6 = 18\). Then braces: \(5 + 18 = 23\). Finally, subtraction: \(23 - 7 = 16\).
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Evaluate \( \{ 2 + [3 \times (4 + 1)] \} \div 5 \).
A · 3
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Simplify \( 3 + \frac{5}{2} \times 4 - 1.5 \) using BODMAS.
A · 11.5
First multiplication: \(\frac{5}{2} \times 4 = 10\). Then addition and subtraction: \(3 + 10 - 1.5 = 11.5\). Correct answer is 11.5, so option A is correct.
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Calculate \( (2.5 + 3.5) \times \frac{4}{5} - 1.2 \).
A · 3.6
Inside parentheses: \(2.5 + 3.5 = 6\). Multiplication: \(6 \times \frac{4}{5} = 4.8\). Subtraction: \(4.8 - 1.2 = 3.6\). Correct answer is 3.6 (option A).
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Identify the common mistake in evaluating \( 8 - 3 + 2 \times 5 \) as \( (8 - 3 + 2) \times 5 = 35 \).
B · Incorrectly grouping terms inside brackets
The mistake is incorrectly grouping \(8 - 3 + 2\) inside brackets before multiplying by 5. According to BODMAS, multiplication should be done before addition and subtraction.
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Which of the following expressions is simplified incorrectly by ignoring BODMAS?
A · \( 6 + 2 \times 3 = 24 \)
Ignoring BODMAS in \(6 + 2 \times 3\) and calculating \( (6 + 2) \times 3 = 24\) is incorrect. The correct calculation is \(6 + (2 \times 3) = 12\).
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Simplify the complex expression \( 3 + 4 \times (2^3 - 5) + \frac{6}{3} \).
A · 17
Calculate inside parentheses: \(2^3 = 8\), then \(8 - 5 = 3\). Multiplication: \(4 \times 3 = 12\). Division: \(6 \div 3 = 2\). Addition: \(3 + 12 + 2 = 17\).
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Evaluate the expression \( \left[ (5 + 3)^2 - 4 \times 3 \right] \div 2 \).
A · 26
Inside parentheses: \(5 + 3 = 8\). Orders: \(8^2 = 64\). Multiplication: \(4 \times 3 = 12\). Subtraction: \(64 - 12 = 52\). Division: \(52 \div 2 = 26\).
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Evaluate the expression:
\[ \frac{(3^2 + 5 \times 4) \div 7 + \sqrt{81}}{2^{3} - 3 \times 2} + \left| -\frac{15}{3} + 4 \right| \times 2 \]
Apply BODMAS strictly and simplify step-by-step.
B · 9
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Evaluate the value of the expression:
\[ \left( \frac{2^{5} - 3^{3}}{7} + 4 \times \sqrt{49} \right) \div \left( 3 + 2 \times (4 - 1)^2 \right) \]
Choose the correct value.
B · 3
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If \( x = 2 \) and \( y = -3 \), evaluate:
\[ \frac{(x^3 - y^3)}{(x - y)} + \sqrt{(x^2 + y^2)} - |xy| \]
Choose the correct value.
B · 19
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Evaluate the expression:
\[ \left( \frac{3 + 5 \times 2^3}{4^2 - 6 \times 5} \right)^2 + \left| 2^3 - 3^2 \right| \]
Choose the correct value.
C · 49
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Evaluate:
\[ \frac{(7 - 3)^3}{3^3 - 2^3} + \sqrt{(5^2 + 12^2)} - \left| 9 - 4 \times 2 \right| \]
Choose the correct answer.
D · 16
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Evaluate the expression:
\[ \left| \frac{2^4 - 3^3}{5} \right| + \sqrt{(3^2 + 4^2)} - 2 \times (3 - 1)^3 \]
Choose the correct value.
B · -2
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Evaluate:
\[ \frac{(2^3 + 3^2)^2}{(5 \times 3) - (4^2 - 7)} + \sqrt{(6^2 - 5^2)} \]
Choose the correct value.
B · 31
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Evaluate:
\[ \frac{(3^3 - 2^4) \times (5 - 2^2)}{7 - (3 + 2)} + | -5 + 3^2 | \]
Choose the correct value.
B · 14
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Evaluate the expression:
\[ \left( 2^{3} + 3^{2} \times 2 \right) \div \left( 5 - 3 \times 1^{2} \right) + \sqrt{(16 - 9)} \]
Choose the correct value.
C · 11
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Evaluate:
\[ \left| 3^3 - 2^4 \right| + \frac{(5^2 - 3^2)}{4} - 2 \times (3 - 1)^2 \]
Choose the correct value.
B · 16
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What is 25% expressed as a decimal?
A · 0.25
25% means 25 per 100, which is \( \frac{25}{100} = 0.25 \).
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If 60 is 30% of a number, what is the number?
A · 200
Let the number be \( x \). Then \( 30\% \times x = 60 \Rightarrow 0.3x = 60 \Rightarrow x = \frac{60}{0.3} = 200 \).
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A price of a product increased from \( \$120 \) to \( \$150 \). What is the percentage increase?
B · 25%
Percentage increase = \( \frac{150 - 120}{120} \times 100 = \frac{30}{120} \times 100 = 25\% \).
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The population of a town decreased from 50,000 to 47,500. What is the percentage decrease?
A · 5%
Percentage decrease = \( \frac{50,000 - 47,500}{50,000} \times 100 = \frac{2,500}{50,000} \times 100 = 5\% \).
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A jacket originally priced at \( \$200 \) is now sold at a 15% discount. What is the selling price?
A · \( \$170 \)
Discount = 15% of 200 = \( 0.15 \times 200 = 30 \). Selling price = \( 200 - 30 = 170 \).
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Convert the fraction \( \frac{3}{5} \) to a percentage.
A · 60%
\( \frac{3}{5} = 0.6 = 60\% \).
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Express 0.375 as a percentage.
A · 37.5%
To convert decimal to percentage, multiply by 100: \( 0.375 \times 100 = 37.5\% \).
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If a car's value depreciates by 10% annually, what will be its value after 2 years if its current value is \( \$20,000 \)?
A · \( \$16,200 \)
Value after 1 year = \( 20,000 \times 0.9 = 18,000 \).Value after 2 years = \( 18,000 \times 0.9 = 16,200 \).
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A shopkeeper gives successive discounts of 10% and 20% on a product. What is the net discount percentage?
B · 28%
Net discount = \( 10\% + 20\% - \frac{10 \times 20}{100} = 30\% - 2\% = 28\% \).
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An article is sold at a profit of 20%. If the cost price is \( \$150 \), what is the selling price?
A · \( \$180 \)
Selling price = Cost price + Profit = \( 150 + 0.20 \times 150 = 150 + 30 = 180 \).
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A trader buys an article for \( \$500 \) and sells it at a loss of 12%. What is the selling price?
A · \( \$440 \)
Loss = 12% of 500 = \( 0.12 \times 500 = 60 \). Selling price = \( 500 - 60 = 440 \).
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If the price of a commodity is increased by 10% and then decreased by 20%, what is the net percentage change in price?
C · 12% decrease
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What is the ratio of 12 to 18 in simplest form?
A · 2:3
Divide both terms by their greatest common divisor, which is 6. So, 12\div6 = 2 and 18\div6 = 3, giving the ratio 2:3.
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If the ratio of boys to girls in a class is 3:4, what fraction of the class are girls?
B · \( \frac{4}{7} \)
Total parts = 3 + 4 = 7. Girls are 4 parts, so fraction is \( \frac{4}{7} \).
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Which of the following represents the ratio of 5 liters to 2 kilograms?
A · 5:2
Ratio compares quantities of the same kind or different units as given. Here, 5 liters to 2 kilograms is 5:2.
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If the ratio of two numbers is 7:5 and their sum is 48, what is the larger number?
A · 28
Let numbers be 7x and 5x. 7x + 5x = 48 \Rightarrow 12x = 48 \Rightarrow x = 4. Larger number = 7 \times 4 = 28.
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Simplify the ratio 48:60.
A · 4:5
GCD of 48 and 60 is 12. Dividing both by 12 gives 4:5.
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Which of the following is the simplest form of the ratio 150:210?
A · 5:7
GCD of 150 and 210 is 30. Dividing both by 30 gives 5:7.
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Simplify the ratio 81:27 to its lowest terms.
A · 3:1
GCD of 81 and 27 is 27. Dividing both by 27 gives 3:1.
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If the ratio of length to width of a rectangle is 5:3, what is the ratio of its width to length?
A · 3:5
The ratio width:length is the reciprocal of length:width, so 3:5.
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Which ratio is equivalent to 4:9?
A · 8:18
Multiplying both terms of 4:9 by 2 gives 8:18, which is equivalent.
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Which of the following ratios is NOT equivalent to 7:10?
D · 28:25
28:25 is not equivalent because 28/25 \( eq \) 7/10.
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If \( \frac{a}{b} = \frac{6}{9} \), which of the following ratios is equivalent to \( \frac{a}{b} \)?
A · 2:3
6:9 simplifies to 2:3 by dividing both terms by 3.
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Find the missing term in the proportion: 5:15 = ? : 45.
D · 15
Using cross multiplication: 5 \times 45 = 15 \times x \Rightarrow 225 = 15x \Rightarrow x = 15.
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If \( \frac{x}{6} = \frac{4}{9} \), what is the value of \( x \)?
A · \( \frac{8}{3} \)
Cross multiply: 9x = 24 \Rightarrow x = \frac{24}{9} = \frac{8}{3}.
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Which of the following is a property of proportions?
A · Product of means equals product of extremes
In a proportion \( \frac{a}{b} = \frac{c}{d} \), product of means (b and c) equals product of extremes (a and d).
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If \( \frac{3}{x} = \frac{6}{10} \), find \( x \).
A · 5
Cross multiply: 3 \times 10 = 6 \times x \Rightarrow 30 = 6x \Rightarrow x = 5.
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If \( \frac{a}{b} = \frac{c}{d} \), which of the following is true?
A · \( a \times d = b \times c \)
The fundamental property of proportion is that the product of extremes equals the product of means.
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A recipe requires ingredients in the ratio 2:3:5. If the total quantity is 100 grams, how much of the second ingredient is needed?
A · 30 grams
Total parts = 2 + 3 + 5 = 10. Second ingredient = \( \frac{3}{10} \times 100 = 30 \) grams.
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If 5 pens cost $15, how much will 8 pens cost at the same rate?
A · $24
Cost per pen = 15/5 = $3. For 8 pens, cost = 8 \times 3 = $24.
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A map uses a scale of 1 cm : 5 km. What is the actual distance represented by 7 cm on the map?
A · 35 km
Actual distance = 7 cm \times 5 km/cm = 35 km.
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If \( \frac{a}{b} = \frac{3}{4} \) and \( a + b = 28 \), find the values of \( a \) and \( b \).
A · 12 and 16
Let \( a = 3x \), \( b = 4x \). Then, 3x + 4x = 28 \Rightarrow 7x = 28 \Rightarrow x = 4. So, a = 12, b = 16.
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If \( x:y = 2:3 \) and \( y:z = 4:5 \), find the ratio \( x:y:z \).
A · 8:12:15
Make y common: \( x:y = 2:3 = 8:12 \), \( y:z = 4:5 = 12:15 \). So, \( x:y:z = 8:12:15 \).
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If \( y \) is directly proportional to \( x \) and \( y = 15 \) when \( x = 3 \), what is \( y \) when \( x = 5 \)?
A · 25
Direct proportion: \( y = kx \). Find \( k = \frac{y}{x} = \frac{15}{3} = 5 \). So, \( y = 5 \times 5 = 25 \).
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If \( y \) is inversely proportional to \( x \) and \( y = 8 \) when \( x = 6 \), find \( y \) when \( x = 12 \).
A · 4
Inverse proportion: \( xy = k \). Find \( k = 8 \times 6 = 48 \). So, \( y = \frac{48}{12} = 4 \).
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If \( y \) varies directly as \( x \) and inversely as \( z \), which of the following expresses \( y \)?
A · \( y = k \frac{x}{z} \)
Directly proportional to \( x \) and inversely proportional to \( z \) means \( y = k \frac{x}{z} \).
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If \( y \) is inversely proportional to \( x^2 \) and \( y = 9 \) when \( x = 2 \), find \( y \) when \( x = 3 \).
A · 4
Inverse proportion: \( yx^2 = k \). Find \( k = 9 \times 4 = 36 \). So, \( y = \frac{36}{9} = 4 \) when \( x=3 \).
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A car travels 60 km in 2 hours. How far will it travel in 5 hours at the same speed?
A · 150 km
Speed = 60/2 = 30 km/h. Distance in 5 hours = 30 \times 5 = 150 km.
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If 3 workers can complete a job in 12 days, how many days will 6 workers take to complete the same job, assuming all work at the same rate?
A · 6 days
Workers and days are inversely proportional. \( 3 \times 12 = 6 \times x \Rightarrow x = 6 \) days.
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A recipe calls for 2 cups of sugar for every 5 cups of flour. How much sugar is needed for 20 cups of flour?
A · 8 cups
Sugar:flour = 2:5. For 20 cups flour, sugar = \( \frac{2}{5} \times 20 = 8 \) cups.
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If the ratio of men to women in a company is 7:5 and there are 60 women, how many men are there?
A · 84
Men:women = 7:5. For 60 women, men = \( \frac{7}{5} \times 60 = 84 \).
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If the cost of 3 kg of apples is $15 and the cost of 5 kg of oranges is $25, what is the compound ratio of the cost of apples to oranges?
D · 45:125
Compound ratio = (3 kg \times 15) : (5 kg \times 25) = 45 : 125.
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Find the compound ratio of 2:3 and 4:5.
A · 8:15
Multiply corresponding terms: 2 \times 4 : 3 \times 5 = 8:15.
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If the compound ratio of 3:4, 5:6, and 7:8 is expressed in simplest form, what is it?
A · 105:153
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In continued proportion, if \( a:b = b:c = 4:5 \), what is the value of \( \frac{c}{a} \)?
A · \( \frac{25}{16} \)
Since \( a:b = b:c = 4:5 \), then \( b = 5k \), \( a = 4k \), \( c = \frac{5}{4}b = \frac{5}{4} \times 5k = \frac{25}{4}k \). So, \( \frac{c}{a} = \frac{\frac{25}{4}k}{4k} = \frac{25}{16} \).
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If \( a:b = b:c = c:d = 2:3 \), find the ratio \( a:d \).
A · 8:27
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If \( a:b = 3:4 \) and \( b:c = 5:6 \), find \( a:b:c \) in simplest form.
A · 15:20:24
Make \( b \) common: \( a:b = 3:4 = 15:20 \), \( b:c = 5:6 = 20:24 \). So, \( a:b:c = 15:20:24 \).
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What is 25% expressed as a ratio in simplest form?
A · 1:4
25% = \( \frac{25}{100} = \frac{1}{4} \), so ratio is 1:4.
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If \( \frac{a}{b} = 0.2 \), what is the equivalent percentage?
A · 20%
0.2 as a percentage is 0.2 \times 100 = 20%.
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Convert 150% to a ratio in simplest form.
A · 3:2
150% = \( \frac{150}{100} = \frac{3}{2} \), so ratio is 3:2.
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If the ratio of boys to girls is 3:4 and boys constitute 60% of the class, what percentage of the class are girls?
A · 40%
If boys are 60%, girls are 100% - 60% = 40%.
