Simplification Questions and Answers

Answer: B

The value is \(\frac{1}{6}\). Work out the numerator and the denominator separately, taking 'of' first. \[ \begin{aligned} \frac{2}{5}\text{ of }1\frac{1}{4} &= \frac{2}{5}\times\frac{5}{4} \\ &= \frac{1}{2} \end{aligned} \] \[ \begin{aligned} \text{Numerator} &= \frac{1}{2}\times\frac{3}{4}\div\frac{1}{2} \\ &= \frac{3}{4} \end{aligned} \] \[ \begin{aligned} 3\frac{2}{3}\text{ of }\frac{1}{2} &= \frac{11}{6},\quad 5\frac{1}{2}\div\frac{11}{6} \\ &= 3 \end{aligned} \] \[ \begin{aligned} \frac{2}{5}\text{ of }3\frac{3}{4} &= \frac{3}{2},\quad 7\frac{1}{2}\div\frac{3}{2} \\ &= 5 \end{aligned} \] \[ \begin{aligned} \text{Denominator} &= 3 - 3\frac{1}{2} + 5 \\ &= \frac{9}{2} \end{aligned} \] \[ \frac{3/4}{9/2} = \frac{1}{6} \] The other options do not match this value.

Answer: B

The value is 60.

Work from the innermost bracket outward:

\[ \begin{aligned} 6 - 8 - 5 &= -7 \\ 6 - 7 - (-7) &= 6 \\ 10 \times 6 &= 60 \end{aligned} \]

The trap is the double negative: subtracting \(-7\) adds 7. Treating it as subtracting 7 gives \(10 \times (-8)\), not any option, and other sign slips give 70 or 80.

Answer: C

24 seconds is \(\frac{1}{450}\) of 3 hours.

Convert both to seconds first:

\[ \begin{aligned} 3\ \text{hours} &= 3 \times 3600 = 10800\ \text{s} \\ \frac{24}{10800} &= \frac{1}{450} \end{aligned} \]

The distractors come from slips in the units: \(\frac{1}{150}\) is 24 seconds as a fraction of one hour, and \(\frac{1}{300}\) is 36 seconds of 3 hours.

Answer: C

The fraction is -5/12.

Let the fraction be x. Then

\[ \begin{aligned} x + \frac{3}{4} &= \frac{1}{3} \\ x &= \frac{1}{3} - \frac{3}{4} = \frac{4 - 9}{12} = -\frac{5}{12} \end{aligned} \]

Check: \(-5/12 + 9/12 = 4/12 = 1/3\). The positive 5/12 comes from dropping the minus sign, and would give 14/12 when 3/4 is added.

The other two options fail the same check: \(2/1 + 3/4 = 11/4\) and \(4/7 + 3/4 = 37/28\), both greater than 1, while the required sum 1/3 is less than 3/4, so the fraction must be negative.

Answer: C

The expression simplifies to 0.5.

With \(a = 0.9\) and \(b = 0.4\), the numerator is \(a^{3} - b^{3}\) and the denominator is \(a^{2} + b^{2} + ab\). Since \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\), the denominator cancels:

\[ \begin{aligned} \frac{a^{3} - b^{3}}{a^{2} + ab + b^{2}} &= a - b \\ &= 0.9 - 0.4 = 0.5 \end{aligned} \]

1.3 is a + b, which would come from the sum-of-cubes identity used by mistake. 0.9 and 0.4 are just the two numbers themselves.

Answer: A

The answer is 75 students preferred online payment, out of 400 in total.

The online and shop groups together are \(\tfrac{3}{16} + \tfrac{5}{8} = \tfrac{3}{16} + \tfrac{10}{16} = \tfrac{13}{16}\) of the students, so the 75 undecided students are the remaining \(\tfrac{3}{16}\).

\[ \begin{aligned} \tfrac{3}{16}N &= 75 \Rightarrow N = 400 \\ \text{online} &= \tfrac{3}{16} \times 400 = 75 \end{aligned} \]

The online group happens to equal the undecided group, both 3/16 of 400. A total of 300 does not make 3/16 of it equal to 75, and 150 online students would be 3/8 of 400, not 3/16.

Answer: C

The descending order is 4, 3, 2, 1.

Convert each fraction to a decimal:

\[ \begin{aligned} \tfrac{8}{9} &\approx 0.889, \quad \tfrac{5}{6} \approx 0.833 \\ \tfrac{3}{4} &= 0.75, \quad \tfrac{5}{7} \approx 0.714 \end{aligned} \]

So 8/9 > 5/6 > 3/4 > 5/7. Each of these fractions is 1 minus a unit-like gap (1/9, 1/6, 1/4, 2/7), and the smaller the gap, the larger the fraction. The option 1, 2, 3, 4 is the ascending order, and the other two options misplace 3/4 or 5/7.

Answer: B

The value is 6.0.

The numerator has the form \(a^{2} - b^{2}\) with a = 4.359 and b = 1.641, and the denominator is a - b:

\[ \begin{aligned} \frac{a^{2} - b^{2}}{a - b} &= a + b \\ &= 4.359 + 1.641 = 6.0 \end{aligned} \]

No long multiplication is needed: the identity \(a^{2} - b^{2} = (a + b)(a - b)\) cancels the denominator. 6.3, 5.0 and 7.3 come from arithmetic slips; the two decimals are chosen to sum to exactly 6.

Answer: D

Three-tenth of the number is 54.

One third of one fourth is one twelfth, so

\[ \begin{aligned} \tfrac{1}{12}x &= 15 \Rightarrow x = 180 \\ \tfrac{3}{10} \times 180 &= 54 \end{aligned} \]

45 is three times 15, which skips finding the number. 35 and 20 do not follow from 180. A quick check: one fourth of 180 is 45 and one third of 45 is 15, as stated.

Answer: A

The fraction of the three pizzas left is \(\frac{2}{9}\).

Work in ninths of a pizza, since each pizza has 9 pieces. The second pizza's \(\frac{2}{3}\) is \(\frac{6}{9}\).

\[ \begin{aligned} \text{eaten} &= \tfrac{8}{9} + \tfrac{6}{9} + \tfrac{7}{9} \\ &= \tfrac{21}{9} \\ \text{left} &= \tfrac{27}{9} - \tfrac{21}{9} = \tfrac{6}{9} \end{aligned} \]

So 6 of the 27 pieces are left, and as a fraction of the three pizzas that is \(\frac{6}{27} = \frac{2}{9}\).

The left-over amount, 6 pieces, is \(\frac{2}{3}\) of one pizza; the question asks for a fraction of all three, so it must be divided by 3. \(\frac{1}{3}\), \(\frac{2}{7}\) and \(\frac{1}{6}\) do not come from the counts.