Decimals and fractions for AFCAT

Decimal fractions is listed by name in the AFCAT Numerical Ability syllabus, so you will see direct questions on it: arrange these fractions, convert this recurring decimal, simplify this decimal expression. But the bigger reason to master it is that fractions sit inside almost every other question. A candidate who sees 37.5% and thinks "3/8" finishes percentage, profit and interest questions noticeably faster.

This guide covers the conversions worth memorising, the methods for each question type and why they work, and then solves AFCAT-level questions step by step.

Conversions to know by heart

Build the rest from these. 7/8 is 1 − 1/8 = 0.875. 3/16 is 3 × 0.0625 = 0.1875. 2/7 is 2 × 0.142857… = 0.285714…

Comparing fractions

Method 1: convert to decimals. Fine when the fractions are ones you know, or when two decimal places settle it.

Method 2: cross-multiply. To compare a/b and c/d (with positive denominators), compare a × d with b × c. Whichever side is bigger belongs to the bigger fraction. Why it works: multiplying both fractions by the positive number b × d does not change which one is larger, and it clears the denominators.

Method 3: same numerator or same denominator. With equal numerators, the smaller denominator gives the bigger fraction (3/7 > 3/8). With equal denominators, the bigger numerator wins.

Recurring decimals to fractions

Pure recurring (the repeat starts right after the point): put the repeating block over as many 9s as it has digits.

• 0.777… = 7/9
• 0.3636… = 36/99 = 4/11
• 0.142857142857… = 142857/999999 = 1/7

Mixed recurring (some digits before the repeat starts): take all the digits up to the end of the first repeat, subtract the non-repeating part, and divide by as many 9s as there are repeating digits followed by as many 0s as there are non-repeating digits.

• 0.1666… = (16 − 1)/90 = 15/90 = 1/6

Why it works. Let x = 0.1666… Then 10x = 1.666… and 100x = 16.666… Subtracting, 90x = 15, so x = 15/90. The 9s come from the shift that lines up the repeating parts, and the 0s from the shift past the non-repeating digit.

Decimal operations

• Adding and subtracting: line up the decimal points.
• Multiplying: ignore the points, multiply the whole numbers, then give the answer as many decimal places as the two numbers had together. 1.2 × 0.05: 12 × 5 = 60, and 1 + 2 = 3 places, so 0.060 = 0.06.
• Dividing: move the decimal point the same number of places in both numbers until the divisor is a whole number. 0.84 ÷ 0.021 = 840 ÷ 21 = 40. Why it works: multiplying both numbers by the same power of 10 does not change the quotient.

Which fractions terminate?

A fraction in its lowest terms gives a terminating decimal only if its denominator has no prime factors other than 2 and 5. So 7/40 (40 = 2³ × 5) terminates as 0.175, while 7/30 (30 has a 3) recurs.

Reduce the fraction first. 9/75 looks as if it should recur because 75 has a 3, but 9/75 = 3/25 = 0.12, which terminates.

Worked examples

Example 1. Arrange 3/5, 5/8, 7/11 and 2/3 in ascending order.

• Decimals: 3/5 = 0.6; 5/8 = 0.625; 7/11 = 0.6363…; 2/3 = 0.6666…
• Order: 3/5 < 5/8 < 7/11 < 2/3.
• Check the closest pair by cross-multiplying: 5/8 vs 7/11 gives 55 vs 56, so 7/11 is larger.

Example 2. Express 0.2353535… as a fraction.

• Non-repeating part: 2 (one digit). Repeating block: 35 (two digits).
• (235 − 2)/990 = 233/990.
• Check: let x = 0.2353535… Then 1,000x = 235.3535… and 10x = 2.3535…, so 990x = 233.

Example 3. Simplify (0.0625 × 0.8) ÷ 0.005.

• 0.0625 × 0.8: 625 × 8 = 5,000, with 5 decimal places, gives 0.05000 = 0.05.
• 0.05 ÷ 0.005 = 50 ÷ 5 = 10.

Example 4. A candidate spends 1/4 of their monthly salary on rent and 2/5 of the remainder on food. They have ₹9,000 left. What is the salary?

• After rent: 3/4 of the salary remains.
• Food: 2/5 of 3/4 = 3/10 of the salary.
• Left: 3/4 − 3/10 = 15/20 − 6/20 = 9/20 of the salary.
• 9/20 of the salary = ₹9,000, so the salary = ₹20,000.

Example 5. Which of these is a non-terminating decimal: 7/40, 9/75, 11/64, 5/12?

• 7/40: 40 = 2³ × 5, terminates.
• 9/75 = 3/25: terminates.
• 11/64: 64 = 2⁶, terminates.
• 5/12: already in lowest terms, and 12 has a factor 3. 5/12 recurs (0.41666…).

Example 6. Given that 1/3.718 = 0.2689, find 1/0.0003718.

• 0.0003718 is 3.718 divided by 10,000.
• So 1/0.0003718 = 10,000 × (1/3.718) = 10,000 × 0.2689 = 2,689.

Practice set

1. Write 0.0625 as a fraction in lowest terms.
2. Find 2.5 × 0.4.
3. Find 0.9 ÷ 0.03.
4. Which is the smallest: 4/9, 5/11 or 3/7?
5. Write 0.4545… as a fraction.
6. Write 0.1777… as a fraction.
7. A tank is 3/5 full. After 24 litres are used, it is 1/3 full. What is its capacity?
8. Simplify 3.6 × 0.25 ÷ 0.009.

Answers:

1. 1/16. 625/10,000; divide both by 625.
2. 1. 25 × 4 = 100, with two decimal places: 1.00.
3. 30. Shift both points two places: 90 ÷ 3.
4. 3/7. 4/9 = 0.444…, 5/11 = 0.4545…, 3/7 = 0.4285…
5. 5/11. 45/99, divided by 9.
6. 8/45. (17 − 1)/90 = 16/90 = 8/45.
7. 90 litres. 3/5 − 1/3 = 9/15 − 5/15 = 4/15 of the tank is 24 litres, so the tank holds 24 × 15 ÷ 4 = 90.
8. 100. 3.6 × 0.25 = 0.9, and 0.9 ÷ 0.009 = 900 ÷ 9 = 100.

What to do next

• Write the conversion table from memory until you get every row right twice in a row.
• Practise ten recurring-decimal conversions, half pure and half mixed.
• Solve 15 fraction word problems, marking every "of the remainder" before you start.
• Move on to simplification, where these conversions do most of the work, and then percentage.