Percentage Questions and Answers

Answer: C

The ₹5 coins hold 10% of the money.

Let the numbers of coins be \(5n\), \(3n\) and \(13n\). Their values are:

\[ \begin{aligned} \text{Total} &= 5n(1) + 3n(5) + 13n(10) \\ &= 5n + 15n + 130n = 150n \end{aligned} \]

The ₹5 coins are worth \(15n\), so their share is \(15n/150n = 10\%\).

\(14\frac{2}{7}\%\) is \(3/21\), the share of ₹5 coins by number, not by value; that is the trap. 21% and 30% match neither the count nor the value.

Answer: A

The sum is 2 % of xy.

Write each percentage as a fraction of 100:

\[ \begin{aligned} \frac{x}{100}y + \frac{y}{100}x &= \frac{2xy}{100} \\ &= 2\% \text{ of } xy \end{aligned} \]

'2 % of (xy/100)' divides by 100 a second time, giving \(2xy/10^{4}\). 'xy % of 100' equals \(xy\), and '100 % of xy' is also \(xy\); both are 50 times too large. A quick check with \(x = y = 10\): 10% of 10 plus 10% of 10 is 2, and 2% of 100 is 2.

Answer: B

The ratio of males to females among the attendees is 1:1.

Take 100 invited guests: 60 men and 40 women. 80 attended, and all 40 women were among them, so the other 40 attendees were men. That gives 40 men to 40 women, or 1:1.

3:2 is the ratio among those invited, not those who came. 2:3 and 2:1 do not match any count in the problem; the key step is that the women account for exactly half of the 80 who attended.

Answer: D

The correct inference is The number of boys who qualified the test is less than the number of girls who qualified.

Girls account for 60% of the qualified candidates, so boys account for the remaining 40%. Whatever the total number who qualified, 40% of it is less than 60% of it.

The other options fail. Equal numbers qualifying would need a 50:50 split, not 40:60. Boys made up 65% of those who appeared and girls 35%, so the numbers appearing were not equal, and more boys appeared than girls, which is the opposite of option C.

Answer: A

The certain inference is that more than half of the pregnant women received medical care at least once from a doctor.

90% of all the women received care at least once, and 60% of those received it from doctors. As a share of all pregnant women:

\[ 0.90 \times 0.60 = 0.54 = 54\% \]

54% is more than half, so the opposite statement ('less than half') is false. The two 'at most once' options cannot be inferred either way: the survey only says each woman received care at least once, and gives no information on how many times anyone saw a doctor.

Answer: D

The correct statement is that there are errors in the official's statement.

Let household consumption be h and other consumption o before summer. The official's figures, together with the known overall fall of 25%, require

\[ \begin{aligned} 0.8h + 1.7o &= 0.75(h + o) \\ 0.05h + 0.95o &= 0 \end{aligned} \]

Both h and o are positive, so the left side is positive and can never be zero. No positive amounts of household and other consumption can fit: a 20% fall in one part and a 70% rise in the other can never add up to an overall fall of 25%, since even the fall in the first part is only 20%.

The three ratio options therefore describe a situation that cannot exist; none of them makes the equation hold.

Answer: B

The price at store S is ₹ 2250.

Let the price at M be m. Then the price at S is 0.9m, and buying from S costs 0.9m + 150. The saving over M is 100:

\[ \begin{aligned} m - (0.9m + 150) &= 100 \\ 0.1m &= 250 \\ m &= 2500,\ \ 0.9m = 2250 \end{aligned} \]

Check: 2250 + 150 = 2400, which is 100 less than 2500. 2500 is the price at M, not at S; 1750 and 1500 do not satisfy the saving condition.

Answer: D

The value of \(q\) is \(\sqrt{5}\).

The correct answer is \(pq\) and the student's answer is \(p/q\). The percentage error is

\[ \begin{aligned} \frac{pq - p/q}{pq} &= 1 - \frac{1}{q^{2}} = 0.8 \\ q^{2} &= 5,\quad q = \sqrt{5} \end{aligned} \]

Check the others: \(q = 5\) gives an error of 96%, \(q = 2\) gives 75%, and \(q = \sqrt{2}\) gives 50%.

Answer: B

The cost is ₹ 504 and the increase is 20%: ₹ 504; 20%.

\[ \begin{aligned} E &= 40\ \text{W} \times 10\ \text{h} \times 180 = 72\ \text{kWh} \\ \text{cost} &= 72 \times 7 = 504 \end{aligned} \]

Two more hours a night makes 12 hours instead of 10. The cost is proportional to the hours, so it rises by 2/10 = 20% (to ₹ 604.8).

₹ 604.8 is the cost at 12 hours a night, not the original cost, which is the trap in two of the options. 12% and 15% do not follow from the ratio 12/10.

Answer: A

The part present is \(\frac{23}{30}\).

Girls are 3/5 of the class and boys 2/5. Of the girls, \(1 - \tfrac{2}{9} = \tfrac{7}{9}\) are present; of the boys, \(1 - \tfrac{1}{4} = \tfrac{3}{4}\):

\[ \begin{aligned} \tfrac{3}{5} \times \tfrac{7}{9} + \tfrac{2}{5} \times \tfrac{3}{4} &= \tfrac{7}{15} + \tfrac{3}{10} \\ &= \tfrac{14}{30} + \tfrac{9}{30} = \tfrac{23}{30} \end{aligned} \]

7/30 is the part absent, the complement of the answer. 17/25 and 18/49 do not arise from the given fractions.