Question

In a dish, the ratio of apples to pears is 2:11. If there are 66 pears, how many apples are there?

Answer 1

12 apples

Explanation:

Givens

We are given that there are 66 pears in the dish.

We are also told that the ratio of apples to pears is 2:11.

We need to find out how many apples are in the dish.

Solve

To solve this problem, we will set up a proportion and solve for an unknown variable that I will denote as a.

To begin, first set up the initial proportion of apples to pears:

`\displaystyle (2)/(11)`

Then, set this equal to our new proportion that will find the number of apples:

`\displaystyle (2)/(11)=(a)/(66)`

To solve, first, cross-multiply:

`\displaystyle (2)/(11)=(a)/(66)\\\\11* a =11a\\\\2*66=132\\\\11a=132`

Then, simplify by dividing both sides of the equation by 11 to solve for a:

`\displaystyle (11a)/(11)=(132)/(11)\\\\a=12`

We can check our work by seeing if our new proportion will reduce to 2:11.

First, substitute 12 as a in the new proportion:

`\displaystyle (12)/(66)`

Then, simplify by finding the GCF. To do this, list all of the numerator's and denominator's factors:

12: 1, 2, 3, 4, 6, 12

66: 1, 2, 3, 6, 11, 22, 33, 66

Isolate the greatest common factor, which is 6 in this case.

The GCF is 6, so divide both the numerator and the denominator by the GCF (6):

`\displaystyle (12)/(6) =2\\\\\displaystyle (66)/(6)=11\\\\\displaystyle (12)/(66) = (2)/(11)`

This proves that the proportions are equal and that there are indeed 12 apples in the dish along with the 66 pears.

Therefore, there are 12 apples in the dish.