IAS 2019 — CSAT Maths

Correct Option (B)

Let the two-digit natural number be represented as 10x + y, where x is the tens digit and y is the units digit. For a two-digit number, x must be an integer from 1 to 9, and y must be an integer from 0 to 9.

The number formed by reversing its digits is 10y + x.

According to the given information, the ratio of the original number to the reversed number is 4:7:

$$\frac{10x + y}{10y + x} = \frac{4}{7}$$10y+x10x+y​=74​

Cross-multiplying the equation yields:

$$7(10x + y) = 4(10y + x)$$7(10x+y)=4(10y+x)

$$70x + 7y = 40y + 4x$$70x+7y=40y+4x

Rearranging the terms to isolate x and y:

$$70x - 4x = 40y - 7y$$70x−4x=40y−7y

$$66x = 33y$$66x=33y

Dividing both sides by 33 gives the relationship between x and y:

$$x = \frac{33}{66}y$$x=6633​y

$$x = \frac{1}{2}y$$x=21​y

Since x and y are digits, and x must be a non-zero integer (1-9), y must be an even integer from 2 to 8 to ensure x is an integer and within the valid range (1-4).

The possible pairs (x, y) that satisfy this condition are:

• If y = 2, then x = (1/2) * 2 = 1. The number is 12.
• If y = 4, then x = (1/2) * 4 = 2. The number is 24.
• If y = 6, then x = (1/2) * 6 = 3. The number is 36.
• If y = 8, then x = (1/2) * 8 = 4. The number is 48.

There are exactly four such pairs of two-digit numbers that satisfy the given condition.

Incorrect Options:

Options A (5), C (3), and D (2) are incorrect because the mathematical derivation and subsequent enumeration of valid digit pairs demonstrate that there are precisely four such numbers, not five, three, or two.