Question

The sum of LCM and HCF of two numbers is 1260. If their LCM is 900 more than their HCF, find the product of two numbers?
a) 203400
b) 194400
c) 198400
d) 205400

Answer 1

Hint: Assume least common multiple to be ‘x’ and highest common factor to be ‘y’. Now you have 2 conditions on x, y. By this you can find 2 equations in x, y. Now we can use the substitution method to solve x, y. First look at the substitution method definition carefully and then understand it, try to convert the variable y in terms of ‘x’ from any equation. Now substitute the value of y back into the remaining equation by this you get an equation which has only one variable, it is called a single variable equation. Try to keep all variable terms on the left hand side and all constants on the right hand side. Algebraically find the value of a variable using the value as you know relation with another variable just substitute it to get the value of that variable. The pair of values will be your result, just verify them by substituting into one of the equations.

Complete step-by-step answer:
Substitution Method: The method of solving a system of equations, it works by solving one of the equations for one of the variables to get in terms of another variable then plugging this back into another equation, solve for the other variable by this you can find both the variables. This method is generally used when there are 2 variables, for more variables it will be tough to solve.
First assume LCM of 2 numbers as x, next assume the HCF of the same two numbers as y. First condition given in the question, can be written as:
Sum of LCM, HCF is equal to the value of 1260. By substituting our assumption, we get the equation: x + y = 1260………………………(i)
Second condition given in the question, can be written as LCM is 900 more than their HCF. As LCM is more it is better to keep LCM on the left hand side. Thus we keep the HCF on the right hand side for easy representation. By using the condition we get the equation as
x = y + 900…………………..(ii)
We know equation (i) can be written as follows;
x + y = 1260……………….(iii)
By substituting the equation (ii) in (iii) we get it as
y + 900 + y = 1260
By simplifying above equation we get the equation as
2y + 900 = 1260
By subtracting 900 on both sides of equation, we get it as
2y + 900 – 900 = 1260 – 900
By simplifying the above equation we get
2y = 360
By dividing with 2 on both sides, we get the values as
y = 180
By substituting x = 1080, y = 180 in equation (i) we get
180 +1080 = 1260
By simplifying the above we get it as 1260 = 1260. Hence verified.
As x = 1080, the least common multiple is 1080 we know product of two numbers $=\text{LCM}\times \text{HCF}$
By substituting both the values, we get the product as xy. By substituting x, y values we can write product as (180) (1080) = 194400
So, the product of two numbers is given as 194400.

So, the correct answer is “Option b”.

Note: After assuming x, y write the product carefully as we must substitute LCM, HCF which makes the result. Verification of the solution must be done to prove that our result is correct, similarly you can first find x in terms of y and then substitute and continue, anyways you will get the same result because the values of x, y won’t change.