Problems on Numbers Questions and Answers

Answer: B

The fraction is \(\frac{5}{3}\).

Test the condition \(x - \frac{2}{x} = \frac{7}{15}\):

\[ \begin{aligned} \frac{5}{3} - 2 \times \frac{3}{5} &= \frac{5}{3} - \frac{6}{5} \\ &= \frac{25 - 18}{15} = \frac{7}{15} \end{aligned} \]

For \(\frac{3}{5}\) the result is \(\frac{3}{5} - \frac{10}{3}\), negative; for \(\frac{4}{3}\) it is \(\frac{4}{3} - \frac{3}{2} \lt 0\); for \(\frac{3}{4}\) it is also negative. Since \(x\) must exceed twice its reciprocal, it must be greater than \(\sqrt{2}\), and only \(\frac{5}{3}\) is.

Answer: C

60% of the original number is 39.

Let the number be \(10a + b\) with \(a + b = 11\). Its reverse is \(10b + a\).

\[ \begin{aligned} 10a + b + 0.4(10b + a) &= 87.4 \\ 10.4a + 5b &= 87.4 \\ 10.4a + 5(11 - a) &= 87.4 \\ 5.4a &= 32.4, \quad a = 6 \end{aligned} \]

So \(b = 5\), the number is 65, and 60% of 65 is 39. Check: \(65 + 0.4 \times 56 = 65 + 22.4 = 87.4\). 25, 32 and 45 do not come from 65.

Answer: D

The lowest integer is -1.

Let the integers be \(n, n+1, \dots, n+5\).

\[ \begin{aligned} 6n + 15 &= 9 \\ n &= -1 \end{aligned} \]

The integers are -1, 0, 1, 2, 3 and 4, whose sum is 9. Starting at 0 gives 15, at 1 gives 21, and at -2 gives 3.

Answer: A

The number is 47.

Let the number be \(10a + b\) with \(a + b = 11\). Adding 27 reverses the digits:

\[ \begin{aligned} 10a + b + 27 &= 10b + a \\ 9(b - a) &= 27, \quad b - a = 3 \end{aligned} \]

With \(a + b = 11\), \(a = 4\) and \(b = 7\): the number is 47, and \(47 + 27 = 74\). 65, 83 and 92 also have digit sums of 11, but adding 27 does not reverse them.

Answer: A

The number is 74.

Let the tens digit be \(x\) and the units digit \(y\). The difference between the number and its reverse is \(9(x - y)\):

\[ \begin{aligned} 9(x - y) &= 27,\ x - y = 3 \\ x + y &= 11 \\ x = 7,\ y &= 4 \end{aligned} \]

Check: \(74 - 47 = 27\) and \(7 + 4 = 11\). 83 has the right digit sum but \(83 - 38 = 45\); 92 and 65 have digit sums of 11 but differences of 63 and 9.

Answer: B

The number is 280.

Let the number be \(N\). The mistaken answer exceeds the correct one by 150:

\[ \begin{aligned} \tfrac{3}{4}N - \tfrac{3}{14}N &= 150 \\ \tfrac{21 - 6}{28}N &= 150 \\ \tfrac{15}{28}N &= 150 \\ N &= 280 \end{aligned} \]

Check: \(\frac{3}{4}\) of 280 is 210 and \(\frac{3}{14}\) of 280 is 60, and \(210 - 60 = 150\). With 500, 180 or 160 the difference would not be 150; for example \(\frac{15}{28}\) of 500 is about 268.

Answer: D

The sum is 2604.

The second number is 12% of 100, which is 12. Its cube is \(12^{3} = 1728\), and that equals two thirds of the first number:

\[ \begin{aligned} \tfrac{2}{3}x &= 1728 \\ x &= 1728 \times \tfrac{3}{2} = 2592 \\ x + 12 &= 2604 \end{aligned} \]

2640 and 2408 come from slips in the multiplication by 3/2, and 2426 from adding the wrong second number. Only 2604 satisfies both conditions.

Answer: B

The number is 105.

Let the number be \(x\). Two thirds of it exceeds three fifths of it by 7:

\[ \begin{aligned} \frac{2x}{3} - \frac{3x}{5} &= 7 \\ \frac{10x - 9x}{15} &= 7 \\ x &= 105 \end{aligned} \]

Check: \(\frac{2}{3} \times 105 = 70\) and \(\frac{3}{5} \times 105 = 63\), which differ by 7. The other options give differences of 4, 10 and 8.

A quicker route: the difference of the two fractions is \(\frac{2}{3} - \frac{3}{5} = \frac{1}{15}\) of the number, and that equals 7, so the number is \(15 \times 7 = 105\).

Answer: B

The number is 8.

A number added to itself 13 times is the number plus 13 more copies of it, 14 copies in all:

\[ \begin{aligned} n + 13n &= 112 \\ 14n &= 112,\ n = 8 \end{aligned} \]

Check: \(14 \times 8 = 112\). The trap is to use 13 copies, but 112 is not a multiple of 13. 7, 9 and 11 give 98, 126 and 154.

Read literally, adding the number to itself 13 times makes 13 additions to the original, which is why the total is 14 times the number.

Answer: C

The original number is 39.

Let the tens digit be \(a\) and the units digit \(b\). Reversing the digits increases the number, so \(b > a\).

\[ \begin{aligned} (10b + a) - (10a + b) &= 54 \\ 9(b - a) &= 54 \\ b - a &= 6 \\ b + a &= 12 \end{aligned} \]

So \(b = 9\), \(a = 3\), and the number is 39; reversing it gives 93, which is 54 more. 93 is the reversed number, the trap. 75 and 48 have digits summing to 12 but differ from their reversals by 18 and 36.