Mathematical Operations Questions and Answers

Answer: D

The value of p is 6.

Apply the rule \(a \oplus y = a^{2} + 2y\) from the inside out:

\[ \begin{aligned} 5 \oplus p &= 25 + 2p \\ 8 \oplus (25 + 2p) &= 64 + 2(25 + 2p) \\ &= 114 + 4p \end{aligned} \]

Setting this equal to 138:

\[ 114 + 4p = 138 \Rightarrow 4p = 24 \Rightarrow p = 6 \]

Check: \(5 \oplus 6 = 37\) and \(8 \oplus 37 = 64 + 74 = 138\). The other values give 134 (p = 5), 130 (p = 4) and 146 (p = 8).

Answer: D

The answer is 156. Replace the signs by their stated meanings: + means -, × means /, - means + and / means ×. Then 120 - 120 × 10/3 becomes \[ 120 + 120 \div 10 \times 3 \] By BODMAS the division and multiplication come first: \[ 120 \div 10 = 12,\quad 12\times 3 = 36 \] \[ 120 + 36 = 156 \] The values 48, 58 and 128 come from applying some of the changes and not others, for example treating - as a minus again.

Division and multiplication have equal rank and are taken left to right, so 120 / 10 is done first and then multiplied by 3. Multiplying 10 by 3 first would give a different result.

Answer: D

The value is 10. Interchange the signs as instructed: plus and minus swap, and multiplication and division swap. The expression 2 ÷ 4 − 10 × 2 + 3 becomes 2 × 4 + 10 ÷ 2 − 3. Following the usual order of operations: 2 × 4 = 8 and 10 ÷ 2 = 5, so the value is 8 + 5 - 3 = 10. The answers 15, 6 and 0 come from applying only part of the swap or from working left to right.

Answer: A

After interchanging the signs, plus becomes minus, minus becomes plus, times becomes divided by and divided by becomes times. The expression 96 x 12 - 7 / 4 + 13 then reads as shown below.

\[ 96 \div 12 + 7 \times 4 - 13 \]

\[ = 8 + 28 - 13 = 23 \]

The answer is 23. The option 1167 is what one gets by multiplying 96 by 12 and the later terms without changing the signs, and 49 or 1137 come from applying the BODMAS order wrongly to the original expression, which the question does not ask for.

Answer: B

Replace each sign by its changed meaning: + means divide, - means plus, x means minus and the division sign means multiply. The expression 56 + 8 - 42 x 7 / 4 then becomes 56 / 8 + 42 - 7 x 4. By the order of operations the division and the multiplication come first, giving 7 + 42 - 28, which is 21. The answer is 21, option (B). The values 28, 14 and 18 come from using a sign with its usual meaning or working strictly from left to right.

Answer: A

The signs are ÷, +, =, ×, − in order. With them the equation reads 85 ÷ 17 + 47 = 13 × 37 − 429. The left side is 85 ÷ 17 = 5, and 5 + 47 = 52. The right side is 13 × 37 = 481, and 481 − 429 = 52. Both sides are 52, so the equation is balanced.

\[ 85 \div 17 + 47 = 52 = 13 \times 37 - 429 \]

Trying the other sets in the same way, none of them makes the two sides equal, so only the first set balances the equation.

Answer: A

The value is 2373.

Replace each sign by its new meaning: '−' means ÷, '×' means +, '÷' means ×, and '+' means −. The equation 325 − 13 × 99 ÷ 25 + 127 becomes 325 ÷ 13 + 99 × 25 − 127. Working it out: 325 ÷ 13 = 25, 99 × 25 = 2475, so 25 + 2475 − 127 = 2373.

2973, 2627 and 2273 come from mistakes in applying the new signs or the order of operations.

The order of operations matters after the signs are replaced: do the division and multiplication first, and then the addition and subtraction. Here 325 ÷ 13 = 25 and 99 × 25 = 2475 come first, and then 25 + 2475 = 2500 and 2500 − 127 = 2373.

Answer: D

The value is 46.

Interchange + with − and × with ÷ in 20 + 2 − 84 × 12 ÷ 4. This gives 20 − 2 + 84 ÷ 12 × 4. Doing the division and multiplication first, 84 ÷ 12 = 7 and 7 × 4 = 28. Then 20 − 2 + 28 = 46.

270, 274 and 50 result from not interchanging all four signs or from the wrong order of operations.

Work in the order of operations: after the interchange the expression is 20 − 2 + 84 ÷ 12 × 4, so the division and multiplication go first, from left to right. That gives 84 ÷ 12 = 7 and then 7 × 4 = 28, and finally 20 − 2 + 28 = 46.

Answer: B

The value is 164.

Interchange + with ÷ and × with − in 27 + 3 − 15 ÷ 33 ÷ 27 + 9 × 7. It becomes 27 ÷ 3 × 15 + 33 + 27 ÷ 9 − 7. Then 27 ÷ 3 = 9, 9 × 15 = 135, 27 ÷ 9 = 3, so 135 + 33 + 3 − 7 = 164.

165, 189 and 258 come from applying the swaps wrongly.

After the interchange the expression is 27 ÷ 3 × 15 + 33 + 27 ÷ 9 − 7. Divisions and multiplications are done first from the left: 27 ÷ 3 = 9, then 9 × 15 = 135, and 27 ÷ 9 = 3. Then add and subtract: 135 + 33 + 3 − 7 = 164.

Answer: A

Interchanging ÷ and × makes the equation correct. The equation then reads \[ 100 \times 20 + 67 - 64 \div 16 \] \[ = 2000 + 67 - 4 = 2063 \] which equals the right side. Interchanging + and − gives 100 ÷ 20 − 67 + 64 × 16, which is 5 − 67 + 1024 = 962. Interchanging × and − gives 100 ÷ 20 + 67 × 64 − 16, which is 5 + 4288 − 16 = 4277. The pair ÷ and + gives 100 + 20 ÷ 67 − 64 × 16, which is a negative value. Only the first pair gives 2063.