Fractions, decimals and percentages

1. Which is a proper fraction?

Answer B: 7/9

In 7/9 the numerator is smaller than the denominator, so its value lies between 0 and 1. The fraction 9/7 is improper, 1 2/9 is a mixed number, and 18/9 equals 2.

2. Which fraction is equivalent to 18/24?

Answer B: 3/4

The highest common factor of 18 and 24 is 6. Dividing both by 6 gives 18/24 = 3/4. The other fractions have different values.

3. Write 84/126 in its simplest form.

Answer A: 2/3

Dividing numerator and denominator by 42 gives 84/126 = 2/3. Although 14/21 is equivalent, it is not simplest because both parts are still divisible by 7.

4. Convert 3 2/5 to an improper fraction.

Answer D: 17/5

Three wholes contain 3 × 5 = 15 fifths. Adding 2 fifths gives 17 fifths, so 3 2/5 = 17/5.

5. Convert 29/6 to a mixed number.

Answer C: 4 5/6

29 ÷ 6 = 4 remainder 5. Therefore 29/6 = 4 5/6. Reversing gives (4 × 6 + 5)/6 = 29/6.

6. Which fraction equals 0.375?

Answer B: 3/8

0.375 = 375/1000. Dividing numerator and denominator by 125 gives 3/8. A division check confirms 3 ÷ 8 = 0.375.

7. Write 7/20 as a decimal.

Answer B: 0.35

Multiplying numerator and denominator by 5 gives 7/20 = 35/100 = 0.35. The decimal therefore represents thirty-five hundredths.

8. Write 62% as a fraction in simplest form.

Answer A: 31/50

62% = 62/100 because percent means per hundred. Dividing numerator and denominator by 2 gives 31/50, which is in simplest form.

9. The bracket notation 0.(4) means 0.444… . What is the exact fraction?

Answer B: 4/9

If x = 0.444…, then 10x = 4.444… . Subtracting gives 9x = 4, so x = 4/9. The fraction 4/10 equals the terminating decimal 0.4, not the recurring decimal.

10. The bracket notation 0.(27) means 0.272727… . What is the exact fraction?

Answer A: 3/11

Let x = 0.272727… . Then 100x = 27.272727… . Subtracting x gives 99x = 27, so x = 27/99 = 3/11.

11. The bracket notation 0.1(6) means 0.1666…, with 1 as a non-recurring prefix. What is the exact fraction?

Answer B: 1/6

Let x = 0.1666… . Then 100x = 16.666… and 10x = 1.666… . Subtraction gives 90x = 15, so x = 15/90 = 1/6.

12. Which list is in ascending order?

Answer A: 0.58, 3/5, 61%

The three values are 0.58, 3/5 = 0.60 and 61% = 0.61. Since 0.58 < 0.60 < 0.61, the ascending order is 0.58, 3/5, 61%.

13. Calculate 5/12 + 7/18.

Answer C: 29/36

5/12 = 15/36 and 7/18 = 14/36. Adding gives 29/36. Since 29 shares no factor with 36, the answer is already in simplest form.

14. Calculate 4 1/3 − 1 5/6.

Answer B: 2 1/2

4 1/3 = 26/6 and 1 5/6 = 11/6. Their difference is 15/6 = 5/2 = 2 1/2.

15. Calculate (14/15) × (25/28), giving your answer in simplest form.

Answer A: 5/6

Cancel 14/28 to 1/2 and 25/15 to 5/3. The product is (1 × 5)/(3 × 2) = 5/6. The option 35/42 is equivalent but is not in simplest form.

16. Calculate 3 1/2 ÷ 7/8.

Answer C: 4

3 1/2 = 7/2. Dividing by 7/8 means multiplying by 8/7: (7/2) × (8/7) = 8/2 = 4.

17. Calculate 6.04 + 0.875 − 2.9.

Answer B: 4.015

First 6.040 + 0.875 = 6.915. Then 6.915 − 2.900 = 4.015. Aligning place values prevents tenths, hundredths and thousandths from being mixed.

18. Calculate 4.62 ÷ 0.7.

Answer C: 6.6

Multiplying dividend and divisor by 10 gives 46.2 ÷ 7, which equals 6.6. A check is 6.6 × 0.7 = 4.62.

19. Calculate 3/4 + 2 × 5/8.

Answer C: 2

Multiply first: 2 × 5/8 = 10/8 = 5/4. Then 3/4 + 5/4 = 8/4 = 2. Adding before multiplying would ignore the correct order of operations.

20. Find 35% of 240.

Answer C: 84

30% of 240 is 72 and 5% is 12. Their sum is 84. Equivalently, 0.35 × 240 = 84.

21. Express 42 as a percentage of 60.

Answer B: 70%

The reference whole is 60, so calculate (42/60) × 100% = 0.7 × 100% = 70%. A check is that 70% of 60 equals 42.

22. Express 54 kg as a percentage of 45 kg.

Answer C: 120%

The units match, so calculate (54/45) × 100% = 1.2 × 100% = 120%. A result above 100% is appropriate because 54 kg is larger than the 45 kg reference quantity.

23. A strip is divided into 12 equal sections and 9 are shaded. Which set gives the shaded amount in three equivalent forms?

Answer A: 3/4, 0.75, 75%

The shaded fraction is 9/12 = 3/4. Dividing 3 by 4 gives 0.75, and multiplying by 100 gives 75%. Therefore 3/4, 0.75 and 75% are equivalent.

24. Calculate 2.4 − (3/5 ÷ 0.75).

Answer C: 1.6

Inside the brackets, 3/5 = 0.6 and 0.6 ÷ 0.75 = 0.8. Then 2.4 − 0.8 = 1.6. The brackets require the division to be completed before the subtraction.