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# [Smart Math] Algebra Problem 5

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

### Problem

The LCM and HCF of two numbers is 300 and 15 respectively. The difference between the two numbers is equal to the HCF. Find the numbers.

1. 15, 30
2. 60, 75
3. 80, 95
4. 40, 55
5. 90, 05

### The Usual Method

Since the product of two numbers = product of LCM and HCF of two numbers,

$x(x+15)=300\times 15$

$\therefore x^{2}+15x-4500=0$

$\therefore (x+75)(x-60)=0$

$\therefore$x = –75 or x = 60

$\because x\ne$ Negative (as the LCM and the HCF both will be negative)

$\therefore$x = 60 and x + 15 = 60 + 15 = 75

(Ans: 2)

Estimated Time to arrive at the answer = 90 seconds.

### Using Technique

Since the product of two numbers = product of LCM and HCF of two numbers, just multiply the options to see which is = $300\times 15$ = 4500 as follows:

15 x 30 = 450 $\ne$ 4500

60 x 75 = 4500 = 4500 hence this should be the answer. There is no need to check further as there cannot be more than one correct answer.

(Ans: 2)

Estimated Time to arrive at the answer = 10 seconds.
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