Lesson Explainer: Proportions Mathematics • 7th Grade

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 7∢14 is the same as the ratio 21∢π‘₯, we can find the value of π‘₯. One way of doing this is to calculate the proportionality coefficient of the first ratio as π‘˜=147=2.

Then, the proportionality coefficient should be the same in the second ratio, so 2=π‘₯21.

We then multiply this equation through by 21 to get π‘₯=42.

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 π‘˜ 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 π‘˜.

We can find π‘˜, the coefficient of proportionality, by noting that 7π‘˜=14, so π‘˜=147=2. 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 1π‘˜=12. 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 π‘ŽβˆΆπ‘ is the same as the ratio π‘βˆΆπ‘‘, then we say that π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional.

In particular, since π‘Žπ‘=𝑐𝑑, we have π‘Žπ‘‘=𝑏𝑐.

The values π‘Ž, 𝑏, 𝑐, and 𝑑 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 (π‘Ž and 𝑑) are called the extremes and the inner terms (𝑏 and 𝑐) are the means.

So, the equation π‘Žπ‘‘=𝑏𝑐 can also be thought of as β€œthe product of the extremes is equal to the product of the means.”

Let’s 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 π‘₯, then find the value of π‘₯.

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 83=96π‘₯.

We can multiply through by 3π‘₯ to get 8π‘₯=96Γ—3, then we divide through by 8 to find π‘₯: π‘₯=96Γ—38=36.

Let’s 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, π‘Ž and 𝑏 are in the same ratio as 𝑏 and 𝑐. We can apply the same reasoning we did in the above examples to find equations involving these values. For example, we know that π‘Žπ‘=π‘˜ and 𝑏𝑐=π‘˜, so we must have π‘Žπ‘=𝑏𝑐.

We can then multiply this equation through by 𝑏𝑐 to get π‘Žπ‘=𝑏.

Similarly, since π‘Ž and 𝑏 are directly proportional, we have that π‘Ž=π‘˜π‘, and since 𝑏 and 𝑐 are directly proportional, we have that 𝑏=π‘˜π‘. Substituting this expression for 𝑏 into the proportionality equation for π‘Ž gives us π‘Ž=π‘˜(π‘˜π‘)π‘Ž=π‘˜π‘.

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 41=164.

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 π‘Ž, 𝑏, and 𝑐 are in continued proportion, then π‘Žπ‘=𝑏𝑐, which tells us π‘Žπ‘=𝑏.

The middle term 𝑏 is called the middle proportional between π‘Ž and 𝑐; π‘Ž is the first proportional, 𝑏 is the second proportional, and 𝑐 is the third proportional. π‘Ž and 𝑐 are also known as the extremes, and 𝑏 is also known as the mean.

If π‘Ž, 𝑏, 𝑐, and 𝑑 are in continued proportion, then π‘Žπ‘=𝑏𝑐=𝑐𝑑.

π‘Ž and 𝑑 are known as the extremes, and 𝑏 and 𝑐 are the means. We can also label the terms as the first, second, third, and fourth proportionals.

Let’s see some examples of solving problems involving continued proportions.

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

If 𝑏 is the middle proportion between π‘Ž and 𝑐, then which of the following is equal to π‘Ž+𝑏𝑏+π‘οŠ¨οŠ¨οŠ¨οŠ¨?

  1. π‘Žπ‘
  2. π‘π‘Ž
  3. 2π‘Žπ‘
  4. 2π‘π‘Ž

Answer

We start by recalling that saying 𝑏 is the middle proportion between π‘Ž and 𝑐 means that π‘Ž, 𝑏, and 𝑐 are in continued proportion. This means that the ratio between successive terms must be equal. Hence, π‘Žπ‘=𝑏𝑐.

Multiplying the equation through by 𝑏𝑐, we get π‘Žπ‘=𝑏.

We can substitute 𝑏=π‘Žπ‘οŠ¨ into the expression we are given to get π‘Ž+𝑏𝑏+𝑐=π‘Ž+(π‘Žπ‘)(π‘Žπ‘)+𝑐.

The numerator has a shared factor of π‘Ž, and the denominator has a shared factor of 𝑐. Taking these out we get π‘Ž+(π‘Žπ‘)(π‘Žπ‘)+𝑐=π‘Ž(π‘Ž+𝑐)𝑐(π‘Ž+𝑐).

Canceling the shared factor of π‘Ž+𝑐 in the numerator and denominator yields π‘Ž(π‘Ž+𝑐)𝑐(π‘Ž+𝑐)=π‘Ž(π‘Ž+𝑐)𝑐(π‘Ž+𝑐)=π‘Žπ‘.

This is option A.

Example 3: Using Properties of Proportions to Find Equivalent Expressions

If 𝑏 is the middle proportion between π‘Ž and 𝑐, then which of the following is equal to π‘βˆ’49π‘π‘Žβˆ’49π‘οŠ©οŠ©οŠ©οŠ©?

  1. π‘π‘οŠ©οŠ©
  2. π‘π‘οŠ¨οŠ¨
  3. π‘π‘οŠ©οŠ©
  4. π‘π‘οŠ¨οŠ¨

Answer

We start by recalling that saying 𝑏 is the middle proportion between π‘Ž and 𝑐 means that π‘Ž, 𝑏, and 𝑐 are in continued proportion, which means the ratio between successive terms must be equal. Hence, π‘Žπ‘=π‘˜π‘π‘=π‘˜,and for some constant π‘˜.

Rearranging each equation, we get π‘Ž=π‘˜π‘ and 𝑏=π‘˜π‘. We can substitute these expressions into the equation we are given: π‘βˆ’49π‘π‘Žβˆ’49𝑏=(π‘˜π‘)βˆ’49𝑐(π‘˜π‘)βˆ’49𝑏.

Distributing the exponents using the rule (π‘π‘ž)=π‘π‘žοŠοŠοŠ gives us (π‘˜π‘)βˆ’49𝑐(π‘˜π‘)βˆ’49𝑏=π‘˜π‘βˆ’49π‘π‘˜π‘βˆ’49𝑏.

The numerator has a shared factor of π‘οŠ©, and the denominator has a shared factor of π‘οŠ©. So we can take out these factors to give π‘˜π‘βˆ’49π‘π‘˜π‘βˆ’49𝑏=ο€Ύπ‘π‘οŠΓ—ο€Ύπ‘˜βˆ’49π‘˜βˆ’49.

We can now cancel the shared factor of π‘˜βˆ’49 to get ο€Ύπ‘π‘οŠΓ—ο€Ύπ‘˜βˆ’49π‘˜βˆ’49=ο€Ύπ‘π‘οŠ.

Next, we take the exponent out of the expression: 𝑐𝑏=𝑐𝑏.

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 π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional, which of the following equals ο„ž6π‘Žβˆ’9𝑏6π‘βˆ’9π‘‘οŠ¨οŠ¨οŠ¨οŠ¨?

  1. π‘π‘Ž
  2. π‘Žπ‘
  3. π‘Žπ‘‘
  4. 𝑑𝑏

Answer

We start by recalling that saying that π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional is equivalent to saying that π‘ŽβˆΆπ‘ is the same as π‘βˆΆπ‘‘. In particular, their coefficients of proportionality will be equivalent, so π‘Ž=π‘˜π‘π‘=π‘˜π‘‘,and for some constant π‘˜.

We then substitute these into the expression we are given in the question to get ο„ž6π‘Žβˆ’9𝑏6π‘βˆ’9𝑑=ο„‘ο„£ο„ 6(π‘˜π‘)βˆ’9𝑏6(π‘˜π‘‘)βˆ’9𝑑.

Next, we distribute the exponents over the parentheses to get ο„‘ο„£ο„ 6(π‘˜π‘)βˆ’9𝑏6(π‘˜π‘‘)βˆ’9𝑑=ο„ž6π‘˜π‘βˆ’9𝑏6π‘˜π‘‘βˆ’9𝑑.

The two terms in the numerator have a shared factor of π‘οŠ¨, and the two terms in the denominator have a shared factor of π‘‘οŠ¨. Taking these out gives us ο„ž6π‘˜π‘βˆ’9𝑏6π‘˜π‘‘βˆ’9𝑑=ο„Ÿπ‘(6π‘˜βˆ’9)𝑑(6π‘˜βˆ’9).

We can now cancel the shared factor of 6π‘˜βˆ’9 to get ο„Ÿπ‘(6π‘˜βˆ’9)𝑑(6π‘˜βˆ’9)=𝑏6π‘˜βˆ’9𝑑6π‘˜βˆ’9=ο„žπ‘π‘‘.

We can now take the square root of the numerator and denominator separately to get ο„žπ‘π‘‘=βˆšπ‘βˆšπ‘‘=𝑏𝑑.

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 π‘Žπ‘=π‘˜π‘π‘˜π‘‘.

Canceling the shared factor of π‘˜ gives π‘Žπ‘=𝑏𝑑.

Hence, the answer is π‘Žπ‘, 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 π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional, then π‘Ž=π‘˜π‘π‘=π‘˜π‘‘.and

Adding these equations together gives π‘Ž+𝑐=π‘˜π‘+π‘˜π‘‘.

Taking out the factor of π‘˜ on the right-hand side of the equation gives us π‘Ž+𝑐=π‘˜(𝑏+𝑑).

Dividing through by (𝑏+𝑑) then tells us π‘Ž+𝑐𝑏+𝑑=π‘˜.

However, π‘˜ is the coefficient of proportionality, so π‘˜=π‘Žπ‘=𝑐𝑑=π‘Ž+𝑐𝑏+𝑑.

This is an interesting property as it allows us to create equivalent fractions. For example, we know that 12=36, which also tells us that 1, 2, 3, and 6 are proportional. So, this property tells us that 12=36=1+32+6.

We can do the same with subtraction; if we subtract the proportionality equations, we get π‘Žβˆ’π‘=π‘˜π‘βˆ’π‘‘π‘˜.

We then take out the factor of π‘˜ and rearrange to get π‘˜=π‘Žβˆ’π‘π‘βˆ’π‘‘.

Applying this property to our fractions tells us 1βˆ’32βˆ’6=12.

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 π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional, then π‘Žπ‘=𝑐𝑑=π‘Ž+𝑐𝑏+𝑑.

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

Let’s 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 π‘Ž7=𝑏4=𝑐14=6π‘Žβˆ’7𝑏+2𝑐3π‘₯, find the value of π‘₯.

Answer

To answer this question, we first recall that if 𝑀, π‘₯, 𝑦, and 𝑧 are proportional, then 𝑀π‘₯=𝑦𝑧=𝑀+𝑦π‘₯+𝑧.

We note that we are given three equivalent fractions, π‘Ž7=𝑏4=𝑐14, and asked to determine an unknown in the fourth.

We cannot just apply this property directly, otherwise we get π‘Ž+𝑏+𝑐7+4+14=6π‘Žβˆ’7𝑏+2𝑐3π‘₯, which we cannot solve for π‘₯. Instead, let us find equivalent fractions for π‘Ž7, 𝑏4, and 𝑐14 so that their numerators match each term in 6π‘Žβˆ’7𝑏+2𝑐.

We multiply the first fraction by 66 to get π‘Ž7=6π‘Ž42.

We multiply the second fraction by βˆ’7βˆ’7 to get 𝑏4=βˆ’7π‘βˆ’28.

We multiply the third fraction by 22 to get 𝑐14=2𝑐28.

We now have 6π‘Ž42=βˆ’7π‘βˆ’28=2𝑐28.

Applying the property we stated earlier, we have 6π‘Žβˆ’7𝑏+2𝑐42βˆ’28+28=6π‘Žβˆ’7𝑏+2𝑐3π‘₯.

We can simplify the denominator of the left-hand side to get 6π‘Žβˆ’7𝑏+2𝑐42=6π‘Žβˆ’7𝑏+2𝑐3π‘₯.

Since the numerators of both sides of the equation are equal, their denominators must also be equal. So, 42=3π‘₯.

We divide this equation through by 3 to see that π‘₯=14.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • The quantity 𝑦 is said to be directly proportional to the quantity π‘₯ when their ratio is the same in all situations. Therefore, 𝑦π‘₯=π‘˜, where π‘˜ is a constant. This equation can be rewritten in the form 𝑦=π‘˜β‹…π‘₯. The constant π‘˜ is the coefficient of proportionality.
  • If the ratio π‘ŽβˆΆπ‘ is the same as the ratio π‘βˆΆπ‘‘, then we say that π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional. This is equivalent to saying that π‘Žπ‘=𝑐𝑑.
  • If π‘Ž, 𝑏, 𝑐, and 𝑑 are proportional, then
    • the values π‘Ž, 𝑏, 𝑐, and 𝑑 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 (π‘Ž and 𝑑) are called the extremes, and the inner terms (𝑏 and 𝑐) are the means,
    • π‘Žπ‘‘=𝑐𝑏,
    • π‘Žπ‘=𝑐𝑑=π‘Ž+𝑐𝑏+𝑑.
  • 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 π‘Žπ‘=𝑏𝑐.
  • If π‘Ž, 𝑏, and 𝑐 are in continued proportion, then
    • the middle term 𝑏 is called the middle proportional between π‘Ž and 𝑐, π‘Ž is the first proportional, 𝑏 is the second proportional, and 𝑐 is the third proportional. π‘Ž and 𝑐 are also known as the extremes, and 𝑏 is also known as the mean,
    • π‘Žπ‘=π‘οŠ¨.

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