SSC CGL Preparation: Concepts and Practice

Learning outcome

Identify fraction types, move between improper and mixed forms, and create equivalent fractions without changing their value.

Understanding fraction forms

A fraction a/b means a ÷ b, where a and b are integers and b is nonzero. The numerator is a; the denominator is b. We normally write the denominator positive. For nonnegative quantities, the denominator describes equal parts of a whole and the numerator counts those parts. Zero is allowed as a numerator.

For the nonnegative fractions classified here, a proper fraction has numerator smaller than denominator, so its value is below one. An improper fraction has numerator greater than or equal to denominator, so it represents at least one whole. Equality belongs to the improper category.

A mixed form separates whole units from a nonzero proper fractional remainder. With no remainder, write just the whole number. Write the addition explicitly, such as 2 + 3/5, instead of placing numbers side by side. For a negative mixed value, use -(2 + 3/5); the minus applies to the entire quantity, not merely the whole-number part.

Equivalent fractions have exactly the same value. Multiply or divide numerator and denominator by the same nonzero factor. When dividing, choose a common divisor that keeps both results integral. This preserves value because a nonzero factor divided by itself equals 1.

A reduced fraction has no common positive divisor of numerator and denominator other than 1. Dividing both by their greatest common divisor reaches this simplest form.

Worked examples

Example 1: Classify and rewrite 11/4.

It is improper because 11 is greater than 4. Division gives 11 = 4 × 2 + 3: two complete groups and three parts remaining. Therefore, 11/4 = 2 + 3/4. The remainder fraction is proper.

Example 2: Rewrite 3 + 2/7 as one fraction.

Each whole contains seven sevenths, so the whole-number part contributes 3 × 7 = 21 sevenths. Add the remaining two: (21 + 2)/7 = 23/7. Thus, 3 + 2/7 = 23/7.

Example 3: Reduce 18/30 and create another equivalent form.

The greatest common divisor is 6. Dividing both numbers gives 18/30 = 3/5. Now multiply both by 4: 3/5 = 12/20. All three fractions have the same value, but only 3/5 is reduced.

Common mistakes

Adding the same number to numerator and denominator does not generally preserve value. Never multiply only one of them. A denominator cannot be zero. In mixed-form conversion, multiply the whole part by the denominator before adding the numerator.

Practice questions

1. Classify 0/7, 7/7 and 5/8 as proper or improper.
2. Convert 29/6 into an explicit mixed form.
3. Convert 4 + 3/8 into one fraction.
4. Find the integer replacing ? in 15/25 = ?/10, and state the reduced form.

Worked answers

1. By the stated definition, 0/7 and 5/8 are proper. The fraction 7/7 is improper because equal numerator and denominator give one whole.
2. Since 29 = 6 × 4 + 5, we have 29/6 = 4 + 5/6.
3. Convert the wholes first: (4 × 8 + 3)/8 = 35/8.
4. Cancel 5: 15/25 = 3/5. Multiply both numbers by 2: 3/5 = 6/10. The missing integer is 6; the reduced form is 3/5.