Percentage Questions and Answers

Answer: C

The original number is 7000.

A 10% decrease followed by a 10% increase multiplies the number by \(0.9 \times 1.1 = 0.99\), a net loss of 1%:

\[ x - 0.99x = 0.01x = 70 \Rightarrow x = 7000 \]

Check: 7000 falls to 6300, then rises to 6930, which is 70 less than 7000. 700 would give a fall of only 7. The common mistake is to assume the decrease and increase cancel; they do not, because the 10% rise is taken on the smaller number.

Answer: D

The percent increase was 200%.

The fraction of employees with cell phones rose from 20% to 60%. Percent increase compares the rise with the starting value:

\[ \begin{aligned} \text{increase} &= \frac{60 - 20}{20} \times 100 \\ &= 200\% \end{aligned} \]

The common mistake is to quote the difference of 40 percentage points as if it were a percent increase. The fraction tripled, which is a rise of 200%, not 3% or 30%; 20% is just the starting figure.

Answer: C

Arun's 2018 salary was about 15.78 percent greater than Mohan's.

Let Mohan's 2018 salary be M and Arun's A. In 2019 they were equal:

\[ \begin{aligned} 1.10\,M &= 0.95\,A \\ \frac{A}{M} &= \frac{1.10}{0.95} = \frac{22}{19} \\ &\approx 1.1579 \end{aligned} \]

So Arun earned about 15.79% more, which the paper prints, cut at two decimals, as 15·78.

Simply adding the two changes gives 15%, close but not offered. 10·98 is close to Mohan's 10% rise alone and ignores Arun's 5% fall; 16·91 and 13·34 do not fit the equation above.

Answer: D

The maximum marks are 300.

Let the maximum be M. The pass mark is 45 more than 20% of M, and also 30 less than 45% of M:

\[ \begin{aligned} 0.20M + 45 &= 0.45M - 30 \\ 0.25M &= 75 \\ M &= 300 \end{aligned} \]

Check: the pass mark is 105. The first candidate scores 60 and falls short by 45; the second scores 135, which is 30 above the pass mark.

With 100, 200 or 400 the two descriptions of the pass mark disagree; for example 400 gives 125 from the first and 150 from the second.

Answer: A

Tuesday's production was 500 shirts.

Let Monday's output be M and Tuesday's T, with \(M + T = 900\). Equal defects means \(0.05M = 0.04T\), so \(M = 0.8T\):

\[ \begin{aligned} 0.8T + T &= 900 \\ T &= 500,\quad M = 400 \end{aligned} \]

Check: 5% of 400 = 20 and 4% of 500 = 20. 400 is Monday's figure, the trap for anyone who mixes up the days.

Shortcut: equal defects means the outputs are in the inverse ratio of the defect rates, M : T = 4 : 5, so Tuesday's share is \(\frac{5}{9} \times 900 = 500\).

Answer: A

The difference is 3200 rupees.

\[ \begin{aligned} \text{mutual funds} &= 28\% \times 40000 = 11200 \\ \text{remaining} &= 40000 - 11200 = 28800 \\ \text{groceries} &= \tfrac{4}{8} \times 28800 = 14400 \\ 14400 - 11200 &= 3200 \end{aligned} \]

The ratio 1 : 4 : 3 has 8 parts, so groceries get half the remainder. Splitting the whole salary instead of the remainder gives a different, wrong figure.

Answer: D

None of the above can be concluded with certainty.

Subtract:

\[ P - Q = 0.65B - 0.5B - (0.5A - 0.4A) = 0.15B - 0.1A \]

Its sign depends on the actual values. With A = 10, B = 9: \(1.35 - 1 \gt 0\), so P is greater. With A = 10, B = 2: \(0.3 - 1 \lt 0\), so Q is greater. Knowing only that A is greater than B does not fix the sign.

P = Q only when \(0.15B = 0.1A\), that is when A = 1.5B, which is allowed but not required.

Answer: B

The tax was 11% of the pre-tax spending.

Take the pre-tax spending as 100. The tax on each part is:

\[ \begin{aligned} 10\% \times 40 + 5\% \times 20 + 15\% \times 40 \\ = 4 + 1 + 6 = 11 \end{aligned} \]

So the total tax is 11% of the amount spent excluding taxes. Taking it as a share of the amount including tax gives about 9.9%, and simply averaging 10, 5 and 15 gives 10%.

Answer: B

C obtained 40% of the marks.

The pass mark is 40%, and A gets 10% less than that. B gets \(11\frac{1}{9}\% = \frac{1}{9}\) less than A, and C gets \(41\frac{3}{17}\% = \frac{7}{17}\) less than A and B together:

\[ \begin{aligned} A &= 40 \times 0.9 = 36 \\ B &= 36 \times \tfrac{8}{9} = 32 \\ A + B &= 68 \\ C &= 68 \times \tfrac{10}{17} = 40 \end{aligned} \]

The fractions are chosen to divide exactly: \(11\frac{1}{9}\%\) is one-ninth, and \(41\frac{3}{17}\%\) is \(\frac{700}{17}\%\), or seven-seventeenths. Reading 'A obtains 10% marks less' as 30% would give non-whole answers that match no option.

Answer: C

The present female population is 5,000.

Let the numbers of males and females be M and F. Then \(M + F = 11000\), and after the increases:

\[ \begin{aligned} 1.11M + 1.2F &= 12660 \\ 1.11(11000) + 0.09F &= 12660 \\ 12210 + 0.09F &= 12660 \\ F &= 5000 \end{aligned} \]

Check: 6,000 males rise by 660 and 5,000 females by 1,000, a total rise of 1,660, from 11,000 to 12,660. 6,000 is the number of males, which is the trap.