Question

The H.C.F. of 9/10, 12/25, 18/35, 21/40 is:
A. 3/5
B. 252/5
c. 3/1400
D. 63/700

Answer 1

Hint – We will solve this question by using a formula, which is, H.C.F. of fractions which is equal to H.C.F. of numerator divided by the L.C.M. of denominator, for which first we will find the H.C.F. of the numerators and then the L.C.M. of the denominators.

Complete Step-by-Step solution:
Here the numbers we have are in fraction, i.e., in $\dfrac{p}{q}$form and the numbers are,
$\dfrac{9}{{10}},\dfrac{{12}}{{25}},\dfrac{{18}}{{35}}$ and $\dfrac{{21}}{{40}}$.

Now, we know that,
H.C.F. of fraction = H.C.F. of numerators / L.C.M. of denominators.

Now,
Highest Common Factor (H.C.F.) of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods of finding the H.C.F. of a given set of numbers:
1. Factorization Method
2. Division Method
But here we will use the Factorization Method.
In Factorization Method, express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.
Now,
The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
There are two methods of finding the L.C.M. of a given set of numbers:
1. Factorization Method
2. Division Method
But here we will use the Factorization Method.
In Factorization Method, resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of the highest powers of all the factors.

Now, we will calculate the H.C.F. of these fractions, by using the formula,
H.C.F. of fraction = H.C.F. of numerators / L.C.M. of denominators.
$\therefore $ H.C.F. of fraction $ = \dfrac{{H.C.F\left( {9,12,18,21} \right)}}{{L.C.M\left( {10,25,35,40} \right)}}$

First we will find the H.C.F. of the numerators,
Factors of $9 = 3 \times 3$
Factors of $12 = 3 \times 2 \times 2$
Factors of $18 = 3 \times 3 \times 2$
Factors of $21 = 3 \times 7$
Common factors $ = 3$
$\therefore $ H.F.C. $ = 3$

Now, we will find the L.C.M. of the denominators,
Factors of $10 = 5 \times 2$
Factors of $25 = 5 \times 5$
Factors of $35 = 5 \times 7$
Factors of $40 = 5 \times 2 \times 2 \times 2$
$\therefore L.C.M. = 1400$

Hence, H.C.F. $ = \dfrac{3}{{1400}}$
Thus, option C is the right answer.

Note – The largest or greatest factor common to any two or more given natural numbers is termed as H.C.F. of the given numbers. For solving these kinds of questions one must know the formula and the difference between H.C.F. and L.C.M.