Percentage Questions and Answers

Answer: B

The maximum marks are 75.

Failing by 2 marks with 28 means the pass mark is 30, and that is 40% of the maximum:

\[ \begin{aligned} 0.4M &= 28 + 2 = 30 \\ M &= \frac{30}{0.4} = 75 \end{aligned} \]

Check: 40% of 75 is 30, and 28 is 2 short of it. Using 28 instead of 30 gives 70, which forgets the 2-mark shortfall. 80 and 85 would make the pass mark 32 and 34, so the student would fail by 4 or 6 marks, not 2.

Answer: A

56% as a simple fraction is \(\frac{14}{25}\).

A percentage is a fraction out of 100, so reduce it to lowest terms by dividing by the common factor 4:

\[ 56\% = \frac{56}{100} = \frac{14}{25} \]

Check: \(\frac{13}{20} = 65\%\) and \(\frac{17}{25} = 68\%\), so neither equals 56%, and since \(\frac{14}{25}\) is listed, 'None of these' is wrong.

Answer: C

Madhu's income is 20% less than Rabi's.

Take Madhu's income as 100. Rabi earns 25% more, so 125. The gap is 25, but it must be measured as a percentage of Rabi's income, since the question compares Madhu with Rabi:

\[ \frac{25}{125} \times 100 = 20\% \]

The trap is to answer 25%, which takes the gap as a share of Madhu's income. When one amount is \(r\%\) more than another, the second is \(\frac{r}{100 + r} \times 100\%\) less than the first.

Answer: D

The ratio is 35:103.

Work out each investment from Abhinav's Rs. 6,000:

\[ \begin{aligned} \text{Sunil} &= 6000 \times 0.70 = 4200 \\ \text{Rita} &= 4200 \times 1.25 = 5250 \\ \text{total} &= 6000 + 4200 + 5250 = 15450 \end{aligned} \]

Rita's share against the total is \(5250 : 15450\). Dividing both by 150 gives 35 : 103. The option 35:104 is one off in the second term, the trap for an arithmetic slip, and the other two ratios do not come from these amounts.

Answer: C

The area shows a 16% decrease.

The area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\), so it changes by the product of the two factors:

\[ \begin{aligned} \text{new area} &= 1.40 \times 0.60 = 0.84 \\ \text{change} &= 0.84 - 1 = -0.16 \end{aligned} \]

That is a 16% decrease. A +40% and a −40% change do not cancel, because the 40% decrease is taken of a larger product. The shortcut \(a + b + \frac{ab}{100} = 40 - 40 - 16 = -16\) gives the same result, which rules out the 20% and 12% options.

Answer: C

B's income is less than A's by 20%.

Take B's income as 100. Then A's income is 125, and the difference of 25 must be measured against A's income:

\[ \frac{25}{125} \times 100 = 20\% \]

The trap is to answer 25%, measuring against B instead of A, but that is not offered. \(16\frac{2}{3}\%\) is the answer for a 20% excess, not a 25% one, and 12% does not follow.

The general rule: if A is r% more than B, then B is less than A by \(\frac{r}{100 + r} \times 100\%\), here \(\frac{25}{125} \times 100 = 20\%\).

Answer: D

The original bill was Rs. 125.

If 20% is deducted, 80% of the bill remains, and that is Rs. 100:

\[ \begin{aligned} 0.8 \times \text{bill} &= 100 \\ \text{bill} &= 125 \end{aligned} \]

Check: 20% of 125 is 25, and 125 - 25 = 100. Rs. 120 comes from adding 20% to 100, which measures the deduction against the wrong amount.

In general, if x% is deducted and R is left to pay, the original amount is \(R \times \frac{100}{100 - x}\), here \(100 \times \frac{100}{80} = 125\).

Answer: A

42 kg is 12.5% of 336 kg.

\[ \frac{42}{336} \times 100 = \frac{1}{8} \times 100 = 12.5\% \]

Since \(336 = 8 \times 42\), 42 is exactly one eighth of 336, which is 12.5%. The other percentages do not match this ratio: 10.5% of 336 is about 35.3 kg and 13% is about 43.7 kg.

Check: 12.5% of 336 is \(336 \div 8 = 42\) kg, exactly the given amount.

Answer: A

The total number of students is 1200.

If 65% passed, 35% failed, and those 35% are the 420 failures.

\[ \begin{aligned} 35\% \text{ of } N &= 420 \\ N &= \frac{420 \times 100}{35} \\ &= 1200 \end{aligned} \]

Check: 35% of 1200 is 420 and 65% is 780, which add to 1200. 1400 is what you get by treating 420 as 30%, and 1250 and 1000 fit no step of the working.

Answer: A

The maximum marks are 1000.

Rohan's score tells us the pass mark: he got 320 and fell 30 short, so the pass mark is 350. Akash got 30% of the maximum M and fell 50 short of the same pass mark:

\[ \begin{aligned} 0.30M + 50 &= 350 \\ 0.30M &= 300 \\ M &= 1000 \end{aligned} \]

Check: 30% of 1000 is 300, which is 50 below 350. With 1200 or 900 the two candidates' figures would give different pass marks, so they cannot be right.