Mathematical Operations Questions and Answers

Answer: A

The value is 22.

Apply the rule a * b = 2a - 3b + ab to each part, taking care that the operation is not symmetric:

\[ \begin{aligned} 3 * 5 &= 6 - 15 + 15 = 6 \\ 5 * 3 &= 10 - 9 + 15 = 16 \\ 6 + 16 &= 22 \end{aligned} \]

Treating 5 * 3 as equal to 3 * 5 would give 12, which is not offered, and the other options come from sign or arithmetic slips.

Answer: B

The value is \(9\tfrac{1}{3}\).

Replace each sign by its meaning: '+' becomes '÷', '-' becomes '×' and '×' becomes '+'. The expression becomes

\[ 12 \div 6 \div 3 \times 2 + 8 \]

Working left to right for division and multiplication, then adding:

\[ \begin{aligned} 12 \div 6 &= 2,\quad 2 \div 3 = \tfrac{2}{3} \\ \tfrac{2}{3} \times 2 &= \tfrac{4}{3} \\ \tfrac{4}{3} + 8 &= 9\tfrac{1}{3} \end{aligned} \]

\(8\tfrac{1}{3}\) is one less than the correct value, and none of the other options results from applying the substitutions and then the usual order of operations.

Answer: D

The correct equation is 24 \(\gt\) 4 × 20 \(\lt\) 2 # 16.

Replace each symbol: \(\gt\) becomes ÷, \(\lt\) becomes +, × becomes =, and # becomes −. The last option then reads

\[ \begin{aligned} 24 \div 4 &= 20 + 2 - 16 \\ 6 &= 6 \end{aligned} \]

which is true. The others fail:

• 24 = 4 ÷ 20 + 2 − 16 gives 24 = −13.8.
• 24 + 4 ÷ 20 − 2 = 16 gives 22.2 = 16.
• 24 − 4 + 20 = 2 ÷ 16 gives 40 = 0.125.

Answer: D

The correct option is 18M 40P8N6S4 = 44.

Replace the letters (M = +, N = ×, P = ÷, S = -) and apply division and multiplication before addition and subtraction (BODMAS):

\[ \begin{aligned} 18 + 40 \div 8 \times 6 - 4 &= 18 + 5 \times 6 - 4 \\ &= 18 + 30 - 4 = 44 \end{aligned} \]

The others fail: A gives \(18 \times 40 \div 8 - 6 + 4 = 88\), not 64; B gives about 52.7, not 55; C gives about -31.3, not 91.

Answer: A

The value is 5.

Replace each letter with its operation: N is \(\times\), P is \(\div\), L is \(+\) and M is \(-\).

\[ 5 \times 5 \div 5 + 5 - 5 \]

Do multiplication and division first, from left to right, then addition and subtraction:

\[ \begin{aligned} 5 \times 5 \div 5 &= 25 \div 5 = 5 \\ 5 + 5 - 5 &= 5 \end{aligned} \]

10 is the value you reach if you stop before the final subtraction of 5. 0 and 15 do not follow from applying the operations in the correct order.

Answer: A

The value is 48.

Replace each symbol by its meaning: - becomes multiplication, + becomes minus, × becomes division and ÷ becomes plus.

\[ \begin{aligned} 15 - 3 + 10 \times 5 \div 5 &\to 15 \times 3 - 10 \div 5 + 5 \\ &= 45 - 2 + 5 = 48 \end{aligned} \]

Working left to right without the usual order of operations gives other values such as 22 or 35, which is how the distractors arise.

Answer: C

The answer is 42.

In each example the result is (first number minus 1) times the second number.

\[ \begin{aligned} 5 \times 4 &\to (5-1) \times 4 = 16 \\ 7 \times 8 &\to (7-1) \times 8 = 48 \\ 6 \times 5 &\to (6-1) \times 5 = 25 \\ 8 \times 6 &\to (8-1) \times 6 = 42 \end{aligned} \]

Thirty-two would come from (8-2) times a different pattern and does not fit the three examples; 29 fits none of them, and the rule gives a definite value, so 'None of the above' is wrong.

Answer: C

The sum of the digits is 28.

Number the places of 76835 from the left: 7 (1st), 6 (2nd), 8 (3rd), 3 (4th), 5 (5th). The odd places hold 7, 8 and 5, which each decrease by 1 to 6, 7 and 4. The even places hold 6 and 3, which each increase by 1 to 7 and 4.

The new number is 6 7 7 4 4, and its digits add up to

\[ 6 + 7 + 7 + 4 + 4 = 28 \]

A quicker check: three digits fall by 1 and two rise by 1, so the original sum of 29 falls by 1 to 28. 29 is the unchanged sum, and 26 comes from decreasing every digit.

Answer: C

Correct answer: 35

Each second number is two and a half times the first: \( 12 \times 2.5 = 30 \), so \[ 14 \times 2.5 = \mathbf{35} \]

Answer: C

Correct answer: 8 R 8 P 8 S 8 Q 8 = 57

Substituting gives \[ 8 \times 8 + 8 \div 8 - 8 = 64 + 1 - 8 = 57. \] The multiplication and the division are evaluated before the addition and subtraction, which is what makes the middle term contribute 1 rather than being folded into a running total.