SSC CGL Preparation: Concepts and Practice

Learning outcome

Convert fractions and decimals in both directions, recover fractions from recurring decimals, and distinguish exact conversions from rounded approximations.

Understanding conversion

A fraction and a decimal can represent the same quantity. Choose a form that makes the task easier without changing its value.

For a terminating decimal, remove the decimal point to obtain the numerator. Use denominator 10, 100, 1000 and so on, according to the number of decimal places. Keep any negative sign and reduce the fraction. Zeros immediately after the decimal point still count as places.

To convert a fraction to a decimal, divide numerator by denominator. When convenient, make an equivalent fraction with a power-of-ten denominator instead. The earlier reduced-denominator test predicts whether an exact terminating form is possible.

For a recurring decimal, let x name the exact number. Multiplication by powers of ten shifts the decimal point. Choose shifts that align identical recurring tails, then subtract. The tails cancel because the complete infinite patterns match, not because a displayed finite prefix is exact.

Use = for exact relationships. When a rounded value differs from the original number, use ≈ and state the precision. Avoid rounding intermediate values before completing a calculation.

Worked examples

Example 1: Convert -0.375 into a reduced fraction.

Three decimal places require denominator 1000: -0.375 = -375/1000. Divide numerator and denominator by 125 to get -3/8. The negative sign remains attached to the whole value; removing the decimal point does not remove the sign.

Example 2: Convert 7/16 into an exact decimal.

Since 16 × 625 = 10000, multiply numerator by the same factor: 7 × 625 = 4375. Thus, 7/16 = 4375/10000 = 0.4375. Four decimal places correspond to the denominator 10000.

Example 3: Convert 0.2777..., where 7 repeats forever, into a fraction.

Let x = 0.2777..., so 10 × x = 2.777... and 100 × x = 27.777..., with identical recurring tails. Subtracting gives 90 × x = 25. Divide by 90: x = 25/90 = 5/18. The initial 2 was nonrepeating, so it required a different shift from a purely recurring decimal.

Common mistakes

Count all decimal positions, including leading zeros. Reduce the final fraction. Do not apply a purely recurring shortcut to a decimal with a nonrepeating beginning. A rounded finite decimal is not generally equal to the original recurring value.

Practice questions

1. Convert 0.048 into a reduced fraction.
2. Convert 1 + 7/20 into an exact decimal.
3. Convert 0.363636..., with recurring block 36, into a reduced fraction.
4. Write the decimal expansion of 5/6 and round it to two decimal places.

Worked answers

1. There are three decimal places: 0.048 = 48/1000. Dividing both numbers by 8 gives 6/125.
2. Since 7/20 = 35/100 = 0.35, the complete quantity is 1 + 0.35 = 1.35.
3. Let x = 0.363636..., so 100 × x = 36.363636..., with matching fractional parts. Subtract x to obtain 99 × x = 36. Hence x = 36/99 = 4/11.
4. Division gives 5/6 = 0.833333..., with 3 recurring. The third decimal digit is 3, so the hundredths digit stays unchanged: 5/6 ≈ 0.83 to two decimal places.