Simplification Questions and Answers

Answer: D

90 packets can be made.

Divide the total salt by the size of one packet:

\[ \begin{aligned} 7\tfrac{1}{2} \div \tfrac{1}{12} &= \tfrac{15}{2} \times 12 \\ &= 15 \times 6 = 90 \end{aligned} \]

Check: each kilogram makes 12 packets, so 7 kg make 84 and the extra half kilogram makes 6 more, 90 in all. 45 is half the answer, as if each kilogram made only 6 packets.

Answer: C

The value is 13.

Work out each bracket first:

\[ \begin{aligned} [(-4) \div 2] \times (-3) &= (-2)(-3) = 6 \\ (-3)[(-3)(-7) - 8] &= (-3)(21 - 8) = -39 \\ (4)[(-48) \div 6] &= 4 \times (-8) = -32 \end{aligned} \]

Then combine them:

\[ 6 - (-39) + (-32) = 6 + 39 - 32 = 13 \]

The trap is the sign of the middle term: subtracting -39 means adding 39. Treating it as -39 gives 6 - 39 - 32 = -65, and slips of the same kind with the other negative signs lead to the wrong options.

Answer: B

The value is 0.

Division comes before subtraction. Since \(5555 \times 222 = 1233210\):

\[ 1233210 \div 5555 - 222 = 222 - 222 = 0 \]

The other options come from slips in the division. Subtracting first, \(5555 - 222\), is not allowed by the order of operations.

Answer: B

The value is 5.

Work out the root, then division and multiplication from left to right, then addition and subtraction (BODMAS):

\[ \begin{aligned} 52 \times 3 \div 4 &= 156 \div 4 = 39 \\ \sqrt{441} \div 7 \times 3 &= 21 \div 7 \times 3 = 9 \\ 32 \div 8 \times 12 &= 4 \times 12 = 48 \\ 39 + 9 + 5 - 48 &= 5 \end{aligned} \]

A common slip is to compute 8 × 12 first in the last term, giving 32 ÷ 96; division and multiplication must be done in order from left to right.

Answer: B

The value is 3.7. Take the square roots first: \[ \sqrt{576} = 24,\quad \sqrt{676} = 26,\quad \sqrt{784} = 28 \] The product is \[ \frac{46}{24}\times\frac{64}{26}\times\frac{22}{28} \] \[ = 1.9167\times 2.4615\times 0.7857 \] \[ \approx 3.707 \] which is 3.7 to one decimal place. The options 4.7, 2.7 and 1.7 differ from the product by a whole unit or more.

Answer: B

The value is 2. Let a = 0.251 and b = 1.749. The numerator is \(a^3 + b^3\) and the denominator is \(a^2 - ab + b^2\). By the identity \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] the fraction reduces to a + b = 0.251 + 1.749 = 2. The other options, 1.498, 0.5 and 2.244, do not result from this simplification. Put a = 0.251 and b = 1.749 so that the numerator is a cubed plus b cubed and the denominator is a squared minus ab plus b squared. Since a + b = 2, the identity gives exactly 2, and no calculator is needed. The answers 1.498 and 2.244 would arise only from careless addition or multiplication.

Answer: B

With x = 81 the powers are x^(1/4) = 3, x^(1/2) = 9 and x^(3/4) = 27. Substituting these into the expression gives the product below.

\[ \frac{81-6}{27+9} \times \frac{9+3}{9+1} \times 3 \]

\[ = \frac{75}{36} \times \frac{12}{10} \times 3 = 7.5 \]

The value is 7.5. The options 8.5, 6.5 and 5.5 come from small arithmetic slips in the powers of 81.

Answer: C

The value is 15. Work left to right with brackets first, treating the number before the curly bracket as a multiplier after the division:

25, 20 and 5 come from grouping 3 and the bracket together as the divisor, which is not the left-to-right order of operations.

The point of the question is the order of operations. Brackets are worked out first, from the innermost, then division and multiplication from left to right. A number placed directly before a bracket, as 3 before the curly bracket, is a multiplier and not a divisor of the whole term, which is why the result is 15 and not the other values.

Answer: B

X = 10000. Divide both sides by 0.00005: \[ X = \frac{0.5}{0.00005} \] \[ = \frac{5 \times 10^{-1}}{5 \times 10^{-5}} = 10^{4} \]

So X is 10000. The option 1000 would give 0.05 on the left side, 1.0000 (which is just 1) would give 0.00005, and 10.000 (which is 10) would give 0.0005, none of which equals 0.5.

Answer: A

The value is 19. Work out each brace. The bar over 1 + 6 groups it as 7, and 14 ÷ 7 ÷ 2 = 1, so the first brace is 7 + 1 = 8. In the second brace, 5 × 2² + 2 = 22 and 2/√16 = 2/4 = 1/2, so it equals 11. The sum is 8 + 11 = 19.

18, 11 and 10 come from dropping the bar or one of the braces.