In this explainer, we will learn how to use properties of proportions to find an unknown value in a proportional relationship and prove algebraic statements.

We say that two or more numbers are in proportion if the ratios of the pairs of numbers are equal. This type of relationship occurs throughout nature and the sciences. For example, the numerators and denominators of equivalent fractions are in proportion.

We can use the proportionality of given pairs of numbers to find missing unknowns. For example, if we are told that the ratio is the same as the ratio , we can find the value of . One way of doing this is to calculate the proportionality coefficient of the first ratio as

Then, the proportionality coefficient should be the same in the second ratio, so

We then multiply this equation through by 21 to get

We can represent this information graphically by using two number lines.

In the first number line, we have the first quantity in each ratio, and in the second number line, we have the second quantity in each ratio. Since the ratios are equal, we multiply by the same constant to convert from the first to the second. We can also add a unit rate to this diagram to determine the rate of conversion of 1 unit; this will be equal to .

We can find , the coefficient of proportionality, by noting that , so . This then allows us to convert 21 by multiplying by 2. It is also worth noting we can convert in the other direction of the arrow using a factor of . Pairs of numbers in this form are called proportional. This an example of saying that the numbers 7, 14, 21, and 42 are in proportion. We can describe this relationship formally as follows.

### Definition: Proportional Numbers

If the ratio is the same as the ratio , then we say that , , , and are proportional.

In particular, since , we have .

The values , , , and are called the terms of the proportion, and we label them as the first proportional, second proportional, third proportional, and fourth proportional. The outer terms ( and ) are called the extremes and the inner terms ( and ) are the means.

So, the equation can also be thought of as βthe product of the extremes is equal to the product of the means.β

Letβs now see another example where we are given three numbers and an unknown in proportion and we need to determine the value of the unknown.

### Example 1: Finding an Unknown Term in a Proportion Using Algebra

If 8 and 3 are in the same ratio as 96 and , then find the value of .

### Answer

We start by recalling that if the ratios of two pairs of numbers are the same, then their proportions are the same. Hence, the quotient of each pair of numbers are equal, giving us

We can multiply through by to get then we divide through by 8 to find :

Letβs now consider a special case where pairs of numbers have the same proportional relationship. Imagine we are told that we have three numbers that are in the same proportion. So, and are in the same ratio as and . We can apply the same reasoning we did in the above examples to find equations involving these values. For example, we know that and , so we must have

We can then multiply this equation through by to get

Similarly, since and are directly proportional, we have that , and since and are directly proportional, we have that . Substituting this expression for into the proportionality equation for gives us

This type of proportionality is called continued proportionality; it can apply to any number of terms, but usually we will see three terms in a continued proportion.

For example, the numbers 1, 4, and 16 are in continued proportion because each successive pair of numbers is in the same ratio. We can calculate these quotients as

We can use this to formally define continued proportion and define a few useful terms to help us describe continued proportions.

### Definition: Continued Proportion

A list of terms is said to be in continued proportion if the ratio between successive terms is constant. Any number of quantities can be in continued proportion. However, we will often be given three or four terms in a continued proportion.

If , , and are in continued proportion, then which tells us

The middle term is called the middle proportional between and ; is the first proportional, is the second proportional, and is the third proportional. and are also known as the extremes, and is also known as the mean.

If , , , and are in continued proportion, then

and are known as the extremes, and and are the means. We can also label the terms as the first, second, third, and fourth proportionals.

Letβs see some examples of solving problems involving continued proportions.

### Example 2: Using Properties of Continued Proportion to Find Equivalent Expressions

If is the middle proportion between and , then which of the following is equal to ?

### Answer

We start by recalling that saying is the middle proportion between and means that , , and are in continued proportion. This means that the ratio between successive terms must be equal. Hence,

Multiplying the equation through by , we get

We can substitute into the expression we are given to get

The numerator has a shared factor of , and the denominator has a shared factor of . Taking these out we get

Canceling the shared factor of in the numerator and denominator yields

This is option A.

### Example 3: Using Properties of Proportions to Find Equivalent Expressions

If is the middle proportion between and , then which of the following is equal to ?

### Answer

We start by recalling that saying is the middle proportion between and means that , , and are in continued proportion, which means the ratio between successive terms must be equal. Hence, for some constant .

Rearranging each equation, we get and . We can substitute these expressions into the equation we are given:

Distributing the exponents using the rule gives us

The numerator has a shared factor of , and the denominator has a shared factor of . So we can take out these factors to give

We can now cancel the shared factor of to get

Next, we take the exponent out of the expression:

This is option C.

We can also use the properties of four numbers in proportion to simplify expressions in the same manner, as we will see in our next example.

### Example 4: Using Properties of Proportions to Find Equivalent Expressions

If , , , and are proportional, which of the following equals ?

### Answer

We start by recalling that saying that , , , and are proportional is equivalent to saying that is the same as . In particular, their coefficients of proportionality will be equivalent, so for some constant .

We then substitute these into the expression we are given in the question to get

Next, we distribute the exponents over the parentheses to get

The two terms in the numerator have a shared factor of , and the two terms in the denominator have a shared factor of . Taking these out gives us

We can now cancel the shared factor of to get

We can now take the square root of the numerator and denominator separately to get

We might note that this is not one of the options given, so we need to rewrite this expression. We can do this by dividing the proportionality equations. We have

Canceling the shared factor of gives

Hence, the answer is , which is option B.

There are many properties of proportions and continued proportions that can be used to simplify expressions involving these concepts. For example, if , , , and are proportional, then

Adding these equations together gives

Taking out the factor of on the right-hand side of the equation gives us

Dividing through by then tells us

However, is the coefficient of proportionality, so

This is an interesting property as it allows us to create equivalent fractions. For example, we know that , which also tells us that 1, 2, 3, and 6 are proportional. So, this property tells us that

We can do the same with subtraction; if we subtract the proportionality equations, we get

We then take out the factor of and rearrange to get

Applying this property to our fractions tells us

We note that these values can be negative, so we only need the additive result. This gives us the following result.

### Property: Proportionality of the Sum

If , , , and are proportional, then

We can just add the numerators and denominators of equivalent fractions separately without affecting their value.

Letβs see an example of using this property to find the value of an unknown.

### Example 5: Using Properties of Proportions to Find Equivalent Expressions

If , find the value of .

### Answer

To answer this question, we first recall that if , , , and are proportional, then

We note that we are given three equivalent fractions, , and asked to determine an unknown in the fourth.

We cannot just apply this property directly, otherwise we get which we cannot solve for . Instead, let us find equivalent fractions for , , and so that their numerators match each term in .

We multiply the first fraction by to get

We multiply the second fraction by to get

We multiply the third fraction by to get

We now have

Applying the property we stated earlier, we have

We can simplify the denominator of the left-hand side to get

Since the numerators of both sides of the equation are equal, their denominators must also be equal. So,

We divide this equation through by 3 to see that

Letβs finish by recapping some of the important points from this explainer.

### Key Points

- The quantity is said to be directly proportional to the quantity when their ratio is the same in all situations. Therefore, , where is a constant. This equation can be rewritten in the form . The constant is the coefficient of proportionality.
- If the ratio is the same as the ratio , then we say that , , , and are proportional. This is equivalent to saying that .
- If , , , and are proportional, then
- the values , , , and are called the terms of the proportion, and we label them as the first proportional, second proportional, third proportional, and fourth proportional. The outer terms ( and ) are called the extremes, and the inner terms ( and ) are the means,
- ,
- .

- A list of terms is said to be in continued proportion if the ratio between successive terms is constant. For three terms, this is equivalent to saying that .
- If , , and are in continued proportion, then
- the middle term is called the middle proportional between and , is the first proportional, is the second proportional, and is the third proportional. and are also known as the extremes, and is also known as the mean,
- .