Finding a Missing Value in a Proportion

Sometimes we will be given only one whole ratio between two terms and a third piece of data that is part of another ratio.

Usually, it will be stated that there is proportionality between the ratios and that we must find the missing data in the ratio.

How are problems solved where a number from the proportion is missing?

We will identify the given ratio in the question between both terms and write it down in the form of a fraction.
We will compare it with the other ratio (also in the form of a fraction) that includes the third piece of data from the question and also the unknown $X$ .
We will solve for $X$ .

Let's look at an example

In the "Freedom for All" zoo, the ratio of camels to peacocks is $2:5$ .

It is known that in the "Firmament" zoo there is the same ratio as in "Freedom for All" between camels and peacocks.

It is also known that, in the Firmament zoo, there are $20$ peacocks.

How many camels are there in the Firmament zoo?

Solution:

1.We will identify the ratio given in the question between both terms and write it down in the form of a fraction.

The question states that the ratio between camels and peacocks is $2:5$ .

We will write it in the form of a fraction and note next to each term what it refers to:

2.We will compare it with the other ratio (also in fraction form) that includes the third piece of data from the question and also the unknown $X$ .

Since it has been stated that the ratio between camels and peacocks is the same in both zoos, we can equate them.

We already know that there are $20$ peacocks in the Firmament zoo, the number of camels is the unknown we need to find.

Therefore, we will draw the following equation:

3.Solve for $X$ . Now that we have an equation with one unknown, we can easily find the $X$ by using a common factor or cross-multiplying.

We will obtain:

$\frac{2}{5}=\frac{X}{20}$

$5X=40$

$X=8$

Let's remember that the idea is the equivalence relation. So, we might ask ourselves, by how much did we multiply $5$ to get to $20$ ?

That is exactly the number by which we will multiply the $2$ to arrive at the required result.

Note, on certain occasions the third piece of data will not be given to us too clearly in the question, so we will also need to express it with an $X$ .

Let's look at another simple example

In a jewelry store, there are $60$ necklaces and $120$ rings.

Maria has $10$ necklaces and, the ratio between the number of necklaces and rings she keeps in her personal jewelry box, is identical to that of the jewelry store.

We must find the number of rings in Maria's jewelry box.

In this question, there is proportionality (equivalence relation) between the jewelry store and Maria's personal jewelry box in the number of items.

Let's write the relation, we will obtain:

$60:120=10:X$