Question

The number of boys and girls in a school are 1168 and 1095 respectively. Express the ratio of the number of boys to that of the girls in the simplest form.

Answer 1

Hint: To find the simplest ratio of the number of boys and girls, we have to break the number of boys and girls into its prime factors and then the ratio of uncommon prime factors will be the simplest form.

Complete step-by-step answer:

As we know that the ratio of two numbers means that both numbers are divided but not completely divided and written in decimal form. Both numbers should be divided till they are only left with different factors.
Like the numbers a and b in the ratio will be expressed as a : b, where a and b does not have any common factors.
Now as we know that if we are asked to write the ratio of number of boys and girls in simplest form.
So, the ratio of number of boys to the number of girls will be \[\dfrac{{{\text{Number of boys}}}}{{{\text{Number of girls}}}} = \dfrac{{1168}}{{1095}}\]. But this is not the simplest ratio because both the numbers can be simplified further.
So, to simply the ration we write both the numerator and denominator into the product of its prime factors.
As we know that the prime factors are those factors of the number that are the smallest prime number by which it is divisible.
So, we can write ratio as \[\dfrac{{1168}}{{1095}} = \dfrac{{2 \times 2 \times 2 \times 2 \times 73}}{{3 \times 5 \times 73}}\]
Now we can see from the above ratio that 73 is a common factor in the numerator and denominator. So, we can remove 73 from the numerator and denominator of the ratio to simplify the ratio.
So, ratio of boys and girls will be \[\dfrac{{2 \times 2 \times 2 \times 2}}{{3 \times 5}} = \dfrac{{16}}{{15}}\] = 16 : 15
Hence, the simplest form of the ratio of the number of boys to the number of girls will be 16 : 15.

Note: Whenever we come up with this type of problem where we are asked to find the simplest form of the ratio of two numbers then we split both the numbers into products of their prime factors and after that we remove the common prime factors of both the numbers. And then the simplest ratio will be the ratio of the product of prime factors of both the numbers that are not common to the prime factors of the other number.