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# [Smart Math] Algebra Problem 17

Here’s and example of a SMART MATH problem for ALGEBRA. ### Problem

50% of a number is equal to 7% of another number. The sum of these numbers is 171. What is the smaller of the two numbers?

1. 17
2. 14
3. 21
4. 28
5. 36

### The Usual Method

Let the smaller of the two numbers be ‘x’. Hence the larger number will be (171 – x).

Also 7% of (171 – x) = 50% of x.

(Note: We are not taking 7% of x and 50% of (171 – x) as x is a smaller number than (171 – x) and 7% of x would be further reduced as compared to 50% of (171 – x). Hence 7% of x << 50% of (171 – x) and not 7% of x = 50% of (171 – x)).

Hence, $\left( 171-x \right)\times 0.07=0.5x$ $\therefore 11.97-0.07x=0.5x$ $\therefore 11.97=0.57x$ $\therefore x=\frac{11.97}{0.57}=21$

(Ans: 3)

Estimated Time to arrive at the answer = 45 seconds.

### Using Technique

Between the two numbers, the smaller one would be the one whose 50% is taken and the larger one would be the one whose 7% is taken. This is discussed in the note above of the usual method.

Hence we need to find the number whose 50% is equal to 7% of some other number. Also note that sum of these two numbers is 171. Hence one of the two numbers is even and the other is odd. $\because 0.5x=0.07y$ (‘y’ = (171 – x)). Thus ‘x’ to be a multiple of 7; the likely values that x can take from the options is 14, 21 and 28. Also note that the other number should be a multiple of 5.

Now if x = 14, y = 171 – 14 (can’t be a multiple of 5, as the last digit will be 7)

If x = 21, y = 171 – 21 = 150 (is a multiple of 5)

If x = 28, y = 171 – 28 (can’t be a multiple of 5, as the last digit will be 3)

(Note: You actually do not even have to subtract 171 – 14 = 167 and 171 – 28 = 143 as just by observing you will know that 171 – 14 and 171 – 28 cannot result in a multiple of 5.)

Hence the answer has to be option ‘3’.

(Ans: 3)

Estimated Time to arrive at the answer = 15 seconds.
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