Simplification: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

This is the square-of-a-difference identity. It comes from expanding (a-b)(a-b) using the distributive rule. That gives a² – ab – ab + b². Combining the two ab terms gives a² – 2ab + b².

This is the square-of-a-sum identity. It comes from expanding (a+b)(a+b) using the distributive rule. That gives a² + ab + ab + b². Combining the two ab terms gives a² + 2ab + b².

This is the difference-of-squares identity: a² – b² = (a+b)(a-b). It works because (a+b)(a-b) expands to a² – ab + ab – b². The middle terms cancel, leaving a² – b². It is useful for quick mental multiplication.

Brackets can be round (), curly {}, or square []. When several are nested together, start with the innermost pair. Solve that pair first, then move one layer outward. Repeat until every bracket is resolved. The shape of the bracket does not change this order.

Division and multiplication rank equally in BODMAS. Neither one is always done before the other. When both appear, work through them strictly left to right, in the order they occur.

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction. These letters give the order of operations. Brackets are always resolved first, before any other step. So brackets come first in every expression.

Let a = 12, b = 3, c = 3, d = 8, and the target = -15. First compute the fixed part: c x d = 3 x 8 = 24. The equation is (a ? b) – 24 = -15. So a ? b must equal -15 + 24 = 9. Testing each operation between a and b, only ‘-‘ works: 12 – 3 = 9. So the missing operation is ‘-‘.

Let a = 12, b = 4, c = 7, d = 9, and the target = -15. First compute the fixed part: c x d = 7 x 9 = 63. The equation is (a ? b) – 63 = -15. So a ? b must equal -15 + 63 = 48. Testing each operation between a and b, only ‘x’ works: 12 x 4 = 48. So the missing operation is ‘x’.

Let a = 3, b = 2, c = 4, d = 8, and the target = -31. First compute the fixed part: c x d = 4 x 8 = 32. The equation is (a ? b) – 32 = -31. So a ? b must equal -31 + 32 = 1. Testing each operation between a and b, only ‘-‘ works: 3 – 2 = 1. So the missing operation is ‘-‘.

Let a = 14, b = 8, c = 6, d = 10, and the target = 52. First compute the fixed part: c x d = 6 x 10 = 60. The equation is (a ? b) – 60 = 52. So a ? b must equal 52 + 60 = 112. Testing each operation between a and b, only ‘x’ works: 14 x 8 = 112. So the missing operation is ‘x’.

Let a = 11, b = 3, c = 6, d = 3, and the target = -4. First compute the fixed part: c x d = 6 x 3 = 18. The equation is (a ? b) – 18 = -4. So a ? b must equal -4 + 18 = 14. Testing each operation between a and b, only ‘+’ works: 11 + 3 = 14. So the missing operation is ‘+’.

Let a = 7, b = 7, c = 6, d = 2, and the target = 2. First compute the fixed part: c x d = 6 x 2 = 12. The equation is (a ? b) – 12 = 2. So a ? b must equal 2 + 12 = 14. Testing each operation between a and b, only ‘+’ works: 7 + 7 = 14. So the missing operation is ‘+’.

Let a = 6, b = 9, c = 6, d = 8, and the target = 6. First compute the fixed part: c x d = 6 x 8 = 48. The equation is (a ? b) – 48 = 6. So a ? b must equal 6 + 48 = 54. Testing each operation between a and b, only ‘x’ works: 6 x 9 = 54. So the missing operation is ‘x’.

Let a = 3, b = 2, c = 9, d = 9, and the target = -80. First compute the fixed part: c x d = 9 x 9 = 81. The equation is (a ? b) – 81 = -80. So a ? b must equal -80 + 81 = 1. Testing each operation between a and b, only ‘-‘ works: 3 – 2 = 1. So the missing operation is ‘-‘.

Let a = 97 = 100 – 3 and b = 95 = 100 – 5. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 100. First, p² = 100² = 10000. Next, p(x+y) = 100 x (3+5) = 100 x 8 = 800. Then, xy = 3 x 5 = 15. Combine: 10000 – 800 + 15 = 9215. So 97 x 95 = 9215.

Let a = 98 = 100 – 2 and b = 91 = 100 – 9. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 100. First, p² = 100² = 10000. Next, p(x+y) = 100 x (2+9) = 100 x 11 = 1100. Then, xy = 2 x 9 = 18. Combine: 10000 – 1100 + 18 = 8918. So 98 x 91 = 8918.

Let a = 997 = 1000 – 3 and b = 999 = 1000 – 1. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 1000. First, p² = 1000² = 1000000. Next, p(x+y) = 1000 x (3+1) = 1000 x 4 = 4000. Then, xy = 3 x 1 = 3. Combine: 1000000 – 4000 + 3 = 996003. So 997 x 999 = 996003.

Let a = 999 = 1000 – 1 and b = 999 = 1000 – 1. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 1000. First, p² = 1000² = 1000000. Next, p(x+y) = 1000 x (1+1) = 1000 x 2 = 2000. Then, xy = 1 x 1 = 1. Combine: 1000000 – 2000 + 1 = 998001. So 999 x 999 = 998001.

Let a = 998 = 1000 – 2 and b = 997 = 1000 – 3. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 1000. First, p² = 1000² = 1000000. Next, p(x+y) = 1000 x (2+3) = 1000 x 5 = 5000. Then, xy = 2 x 3 = 6. Combine: 1000000 – 5000 + 6 = 995006. So 998 x 997 = 995006.

Let a = 96 = 100 – 4 and b = 93 = 100 – 7. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 100. First, p² = 100² = 10000. Next, p(x+y) = 100 x (4+7) = 100 x 11 = 1100. Then, xy = 4 x 7 = 28. Combine: 10000 – 1100 + 28 = 8928. So 96 x 93 = 8928.

Let a = 994 = 1000 – 6 and b = 993 = 1000 – 7. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 1000. First, p² = 1000² = 1000000. Next, p(x+y) = 1000 x (6+7) = 1000 x 13 = 13000. Then, xy = 6 x 7 = 42. Combine: 1000000 – 13000 + 42 = 987042. So 994 x 993 = 987042.

Let a = 99 = 100 – 1 and b = 96 = 100 – 4. Use (p-x)(p-y) = p² – p(x+y) + xy, with p = 100. First, p² = 100² = 10000. Next, p(x+y) = 100 x (1+4) = 100 x 5 = 500. Then, xy = 1 x 4 = 4. Combine: 10000 – 500 + 4 = 9504. So 99 x 96 = 9504.

Let a = 89 and b = 15. The identity is (a-b)² = a² – 2ab + b². First, a² = 89² = 7921. Next, 2ab = 2 x 89 x 15 = 2670. Then, b² = 15² = 225. Combine the terms: 7921 – 2670 + 225 = 5476. So (89-15)² = 5476.

Let a = 47 and b = 13. The identity is (a+b)² = a² + 2ab + b². First, a² = 47² = 2209. Next, 2ab = 2 x 47 x 13 = 1222. Then, b² = 13² = 169. Add the three terms: 2209 + 1222 + 169 = 3600. So (47+13)² = 3600.

Let a = 69 and b = 5. The identity is (a-b)² = a² – 2ab + b². First, a² = 69² = 4761. Next, 2ab = 2 x 69 x 5 = 690. Then, b² = 5² = 25. Combine the terms: 4761 – 690 + 25 = 4096. So (69-5)² = 4096.