Decimals and fractions for RRB Group D

Fractions and decimals are two ways of writing the same number. 3/4 and 0.75 are identical; only the clothes are different. Questions on them are rarely hard, but they are easy to rush. A candidate who can switch between the two forms in a second, and who never loses a decimal place, picks up these marks in under a minute each.

They also turn up inside other topics. Percentage, profit and loss, ratio and simplification all lean on fractions, so time spent here pays off across the whole maths section.

Types of fractions

To turn a mixed number into an improper fraction, multiply the whole number by the bottom and add the top: 2 1/4 = (2 × 4 + 1)/4 = 9/4.

Operations on fractions

For mixed numbers, add the whole numbers and the fractions separately. It keeps the numbers small. If you need the LCM quickly, see our LCM and HCF guide.

Converting between fractions and decimals

Fraction to decimal: divide the top by the bottom. 7/8 = 7 ÷ 8 = 0.875.

Decimal to fraction: write the digits over 1 followed by as many zeros as there are decimal places, then cut down. 0.375 = 375/1000 = 3/8.

Learn these pairs by heart. They save more time than any trick:

Recurring decimals

Some fractions never stop: 1/3 = 0.333… The repeating part is written with a bar in books; here we show it with dots.

• Pure recurring (the repeat starts right after the point): put the repeating digits over the same number of 9s. 0.777… = 7/9. 0.4545… = 45/99 = 5/11.
• Mixed recurring (some digits before the repeat starts): take all the digits minus the non-repeating digits, over as many 9s as repeating digits followed by as many 0s as non-repeating digits. 0.1666… = (16 − 1)/90 = 15/90 = 1/6.

Comparing fractions

Three ways, pick whichever is fastest:

1. Convert to decimals. Good when you know the decimals already.
2. Cross-multiply two at a time. For 4/7 and 5/9: 4 × 9 = 36 and 5 × 7 = 35. Since 36 is bigger, 4/7 is bigger.
3. Same top or same bottom. With the same denominator, the bigger top wins. With the same numerator, the smaller bottom wins: 3/7 is bigger than 3/8.

Decimal operations

• Adding or subtracting: line up the decimal points. 4.5 + 0.35 = 4.85.
• Multiplying: multiply as whole numbers, then count the decimal places in both numbers together. 0.4 × 0.05: 4 × 5 = 20; places 1 + 2 = 3, so 0.020 = 0.02.
• Dividing: move the point in both numbers the same number of places until the divisor is a whole number. 0.36 ÷ 0.012 = 360 ÷ 12 = 30.
• Multiplying or dividing by 10, 100, 1,000: just move the point right or left.

Worked examples

Example 1: Which is largest: 3/5, 5/8 or 7/12?

• 3/5 = 0.6; 5/8 = 0.625; 7/12 ≈ 0.583.
• Largest: 5/8.

Example 2: 0.25 + 0.5 × 0.2

• BODMAS: multiply first. 0.5 × 0.2 = 0.1.
• 0.25 + 0.1 = 0.35.

Example 3: 2 3/4 + 1 5/6 − 1 1/3

• Whole numbers: 2 + 1 − 1 = 2.
• Fractions, LCM 12: 9/12 + 10/12 − 4/12 = 15/12 = 1 1/4.
• Total: 2 + 1 1/4 = 3 1/4.

Example 4: Write 0.2727… as a fraction.

• Pure recurring, two repeating digits: 27/99.
• Cut down by 9: 3/11.

Example 5: 0.36 ÷ 0.012

• Move the point three places in both: 360 ÷ 12 = 30.

Example 6: A worker spends 1/4 of their monthly pay on rent and 2/5 of the remainder on food. ₹9,000 is left. What is the monthly pay?

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

Common mistakes

• Adding fractions by adding tops and bottoms: 1/2 + 1/3 is 5/6, not 2/5.
• Losing a decimal place when multiplying decimals.
• Forgetting BODMAS in mixed expressions with decimals.
• Treating 0.1666… as 16/99. It is mixed recurring, so the answer is 1/6.
• Taking a fraction of the whole when the question says "of the rest".

Practice set

1. 1/2 + 1/3
2. 0.4 × 0.05
3. Write 0.375 as a fraction.
4. Which is largest: 2/3, 3/5 or 5/9?
5. 5/6 − 1/4
6. 1.44 ÷ 0.12
7. Write 0.1333… as a fraction.
8. Find 2/3 of 3/4 of 480.

Answers:

1. 5/6. LCM 6: 3/6 + 2/6.
2. 0.02. 4 × 5 = 20, three decimal places.
3. 3/8. 375/1000 cut down by 125.
4. 2/3. 2/3 ≈ 0.667, 3/5 = 0.6, 5/9 ≈ 0.556.
5. 7/12. LCM 12: 10/12 − 3/12.
6. 12. 144 ÷ 12.
7. 2/15. Mixed recurring: (13 − 1)/90 = 12/90 = 2/15.
8. 240. 3/4 of 480 = 360; 2/3 of 360 = 240.

What to do next

• Learn the fraction–decimal table until you can recite it both ways.
• Practise ten comparisons using cross-multiplication, timing yourself.
• Convert five pure and five mixed recurring decimals, then check by division.
• Revise BODMAS, then move on to percentage, which uses these fractions constantly.