SSC CGL Preparation: Concepts and Practice

Learning outcome

Simplify expressions containing several operations, choose useful shortcuts and translate a short numerical situation into a correctly grouped expression.

Concepts and reasons

A mixed numerical expression combines different operations. Here we use integer values, positive integer exponents and divisions with integer results. The aim is to reduce work without changing what the expression means.

Read its structure before calculating. Identify the complete divisor of each division, the base of each power and the groups affected by subtraction. You may simplify independent groups separately, but preserve the operations connecting them.

The distributive law states a × (b + c) = a × b + a × c. The same quantity a contributes to both parts, so separating or combining their multipliers preserves the total. In reverse, this becomes common-factor extraction. For a difference, a × b - a × c = a × (b - c).

Subtracting a group reverses every term’s sign: a - (b - c) = a - b + c. The subtraction applies to the whole group, not just its first term. Evaluating the bracket first is often safer for a beginner than removing it immediately.

Addition and multiplication permit regrouping; subtraction and division generally do not. Before reordering signed terms, rewrite subtraction as addition of opposites and retain their signs.

Cancellation in division requires a factor of the whole dividend: (a × b) ÷ a = b when a ≠ 0. A matching number in just one term of a sum cannot be cancelled from the whole sum.

Worked examples

Example 1 — Collect a common factor. Simplify 37 × 28 + 37 × 12. Both terms contain factor 37, so the expression equals 37 × (28 + 12) = 37 × 40 = 1480.

Example 2 — Preserve nested signs. Simplify 84 - {19 - [8 - (-5)]} + (-2)^3. First, 8 - (-5) = 13; then 19 - 13 = 6. Also, (-2)^3 = (-2) × (-2) × (-2) = -8. The result is 84 - 6 - 8 = 70.

Example 3 — Combine different strategies. Simplify 240 ÷ [3 × (7 - 3)] + [6 × 18 - 6 × 8] - (-3)^2. The divisor is 3 × 4 = 12. The second group is 6 × (18 - 8) = 60. The square is 9. Therefore the result is 20 + 60 - 9 = 71.

Common mistakes

Do not cancel across addition or subtraction. Do not change only one sign when subtracting an entire group. A negative number squared is positive, but subtracting that square still decreases the total.

Practice questions

1. Simplify 64 × 27 - 64 × 17 using a common factor.
2. Simplify 120 - {35 - [18 - (-7)]}.
3. Simplify 180 ÷ [5 × (9 - 6)] + 7 × (15 - 9) - (-2)^3.
4. A store has 9 boxes containing 24 pens each. It sets aside 36 pens and packs the rest equally into 6 bundles. Write one bracketed expression and find the pens per bundle.

Answers and explanations

1. 640. Both terms contain factor 64, so 64 × (27 - 17) = 64 × 10 = 640.
2. 110. The inner value is 18 - (-7) = 25. Then 35 - 25 = 10, giving 120 - 10 = 110.
3. 62. The divisor is 5 × 3 = 15, so 180 ÷ 15 = 12. The product is 7 × 6 = 42, and (-2)^3 = -8. Thus 12 + 42 - (-8) = 12 + 42 + 8 = 62.
4. 30 pens. The expression is (9 × 24 - 36) ÷ 6. First find 216 - 36 = 180 remaining pens; then 180 ÷ 6 = 30. The brackets ensure that all remaining pens are divided, not only those set aside.