Lesson Video: Proportions Mathematics • 7th Grade

Video Transcript

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

Two or more numbers are said to be in proportion if the ratios of the pairs of numbers are equal. For example, the numerators and denominators of equivalent fractions are in proportion. Let’s imagine we are told that the ratio seven to 14 is the same as the ratio 21 to 𝑥. One way to find the value of 𝑥 is to calculate the proportionality coefficient of the first ratio. This value 𝑘 is equal to 14 divided by seven, which is equal to two. The proportionality coefficient must be the same for the second ratio. So 𝑥 over 21 is equal to two. Multiplying through by 21, we see that 𝑥 is equal to 42. The ratio seven to 14 is equal to the ratio 21 to 42. Let’s now consider a more formal definition of this.

If the ratio 𝑎 to 𝑏 is the same as the ratio 𝑐 to 𝑑, then we say that 𝑎, 𝑏, 𝑐, and 𝑑 are proportional. In particular, since 𝑎 over 𝑏 is equal to 𝑐 over 𝑑, we have 𝑎𝑑 is equal to 𝑏𝑐. The values 𝑎, 𝑏, 𝑐, and 𝑑 are called the terms of the proportion. And we label them as the first proportional, the second proportional, third proportional, and fourth proportional. The outer terms 𝑎 and 𝑑 are called the extremes, and the inner terms 𝑏 and 𝑐 are the means. This means that the equation 𝑎𝑑 is equal to 𝑏𝑐 can be thought of as the product of the extremes is equal to the product of the means. We will now look at an example where we need to use this property.

If eight and three are in the same ratio as 96 and 𝑥, then find the value of 𝑥.

We begin by recalling that if the ratios of two pairs of numbers are the same, then their proportions are the same. This means that the quotient of each pair of numbers are equal. Eight over three is equal to 96 over 𝑥. We can multiply both sides of this equation by three 𝑥. The left-hand side becomes eight 𝑥. And the right-hand side is 96 multiplied by three, which is 288. Our final step to calculate the value of 𝑥 is to divide through by eight. And this is equal to 36. If eight and three are in the same ratio as 96 and 𝑥, the value of 𝑥 is 36.

It is worth noting that in questions like this, we can quote the following result. If 𝑎 over 𝑏 is equal to 𝑐 over 𝑑, then 𝑎𝑑 is equal to 𝑏𝑐. The product of eight and 𝑥 must be equal to the product of three and 96.

We will now move on to consider a list of three or four terms in 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, for the purposes of this video, we will deal with three or four terms.

If we consider three terms 𝑎, 𝑏, and 𝑐 that are in continued proportion, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐. This also means that 𝑎𝑐 is equal to 𝑏 squared, where the middle term 𝑏 is called the middle proportion or mean and 𝑎 and 𝑐 are known as the extremes. If four terms 𝑎, 𝑏, 𝑐, and 𝑑 are in continued proportion, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐, which is equal to 𝑐 over 𝑑, where 𝑎 and 𝑑 are known as the extremes and 𝑏 and 𝑐 are the means. We can also label these as the first, second, third, and fourth proportionals.

Let’s now consider one example of each type.

If 𝑏 is the middle proportion between 𝑎 and 𝑐, then which of the following is equal to 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared? Is it (A) 𝑎 over 𝑐, option (B) 𝑐 over 𝑎, option (C) two 𝑎 over 𝑐, or option (D) two 𝑐 over 𝑎?

We begin by recalling that if three numbers are in continued proportion, where 𝑏 is the middle proportion between 𝑎 and 𝑐, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐. Cross multiplying, this means that 𝑎𝑐 is equal to 𝑏 squared. The expression that we need to simplify in this question is 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared. We will begin by replacing 𝑏 squared with 𝑎𝑐. On the numerator, we have 𝑎 squared plus 𝑎𝑐, and on the denominator, 𝑎𝑐 plus 𝑐 squared.

Our next step is to factor out a common factor of 𝑎 on the numerator and a common factor of 𝑐 on the denominator. This leaves us with 𝑎 multiplied by 𝑎 plus 𝑐 over 𝑐 multiplied by 𝑐 plus 𝑎. As addition is commutative, we can cancel a common factor of 𝑎 plus 𝑐. And this leaves us with a simplified expression of 𝑎 over 𝑐. The correct answer is therefore option (A). If 𝑏 is the middle proportion between 𝑎 and 𝑐, then 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared is equal to 𝑎 over 𝑐.

In our next example, we will use the properties of four numbers in proportion.

If 𝑎, 𝑏, 𝑐, and 𝑑 are proportional, which of the following equals the square root of six 𝑎 squared minus nine 𝑏 squared over six 𝑐 squared minus nine 𝑑 squared? Is it option (A) 𝑎 over 𝑑, option (B) 𝑎 over 𝑐, option (C) 𝑐 over 𝑎, or option (D) 𝑑 over 𝑏?

We begin by recalling that saying that 𝑎, 𝑏, 𝑐, and 𝑑 are proportional is equivalent to saying that the ratio of 𝑎 to 𝑏 is equal to the ratio of 𝑐 to 𝑑. In particular, their coefficients of proportionality will be equivalent. So 𝑎 is equal to 𝑘 multiplied by 𝑏, and 𝑐 is equal to 𝑘 multiplied by 𝑑 for some constant 𝑘. We will begin by substituting these expressions for 𝑎 and 𝑐. This gives us the square root of six multiplied by 𝑘𝑏 all squared minus nine 𝑏 squared over six multiplied by 𝑘𝑑 squared minus nine 𝑑 squared.

The numerator of the fraction under the square root can be rewritten as six 𝑘 squared 𝑏 squared minus nine 𝑏 squared. And the denominator is equal to six 𝑘 squared 𝑑 squared minus nine 𝑑 squared. Next, we can factor out 𝑏 squared from the numerator and 𝑑 squared from the denominator. And after doing this, we can cancel a shared factor of six 𝑘 squared minus nine. Our expression simplifies to the square root of 𝑏 squared over 𝑑 squared. Recalling that we can take the square root of the numerator and denominator separately, this simplifies to 𝑏 over 𝑑.

At this stage, we note that this is not one of the four options. We will therefore consider the two proportionality equations we wrote earlier. Dividing these, we see that 𝑎 over 𝑐 is equal to 𝑘𝑏 over 𝑘𝑑. Since the shared factor 𝑘 is a nonzero constant, we can cancel this, leaving us with 𝑎𝑐 is equal to 𝑏𝑑. We can therefore conclude that the correct answer is option (B). If 𝑎, 𝑏, 𝑐, and 𝑑 are proportional, then the square root of six 𝑎 squared minus nine 𝑏 squared over six 𝑐 squared minus nine 𝑑 squared is equal to 𝑎 over 𝑐.

Before looking at one final example, we will consider one further property of proportionality. The proportionality of the sum states that if 𝑎, 𝑏, 𝑐, and 𝑑 are proportional, then 𝑎 over 𝑏 is equal to 𝑐 over 𝑑, which is equal to 𝑎 plus 𝑐 over 𝑏 plus 𝑑. We can simply add the numerators and denominators of equivalent fractions separately without affecting their value. Let’s now look at an example of this in action.

If 𝑎 over seven is equal to 𝑏 over four which is equal to 𝑐 over 14 which is equal to six 𝑎 minus seven 𝑏 plus two 𝑐 over three 𝑥, find the value of 𝑥.

In order to answer this question, we firstly recall that if four numbers 𝑤, 𝑥, 𝑦, and 𝑧 are proportional, then 𝑤 over 𝑥 is equal to 𝑦 over 𝑧 which is equal to 𝑤 plus 𝑦 over 𝑥 plus 𝑧. In this question, we are given three equivalent fractions: 𝑎 over seven, 𝑏 over four, and 𝑐 over 14. And we are asked to determine an unknown in the fourth.

Unfortunately, we cannot apply the property directly, as we get the equation shown, which we cannot solve for 𝑥. Instead, we will find equivalent fractions for 𝑎 over seven, 𝑏 over four, and 𝑐 over 14, so their numerators match each term on the right-hand side. Multiplying the numerator and denominator of the first fraction by six, we see that 𝑎 over seven is equal to six 𝑎 over 42. In a similar way, 𝑏 over four is equivalent to negative seven 𝑏 over negative 28. And 𝑐 over 14 is equal to two 𝑐 over 28.

We now have six 𝑎 over 42 is equal to negative seven 𝑏 over negative 28 which is equal to two 𝑐 over 28 which is equal to six 𝑎 minus seven 𝑏 plus two 𝑐 over three 𝑥. Applying the property to the first three fractions, we have six 𝑎 minus seven 𝑏 plus two 𝑐 over 42 minus 28 plus 28. And we know this is equal to six 𝑎 minus seven 𝑏 plus two 𝑐 over three 𝑥. Since the numerators of both sides of the equation are equal, their denominators must be equal. So we have three 𝑥 is equal to 42 minus 28 plus 28. This simplifies to three 𝑥 is equal to 42. And dividing through by three, we have 𝑥 is equal to 14.

If 𝑎 over seven is equal to 𝑏 over four which is equal to 𝑐 over 14 which is equal to six 𝑎 minus seven 𝑏 plus two 𝑐 over three 𝑥, then the value of 𝑥 is 14.

We will now summarize the key points from this video. If 𝑦 is directly proportional to 𝑥, then 𝑦 is equal to 𝑘 multiplied by 𝑥, where the constant 𝑘 is known as the coefficient of proportionality. If two ratios 𝑎 to 𝑏 and 𝑐 to 𝑑 are equal, then 𝑎 over 𝑏 is equal to 𝑐 over 𝑑. And cross multiplying, 𝑎 multiplied by 𝑑 is equal to 𝑏 multiplied by 𝑐. Another way of writing this is as follows. If 𝑎, 𝑏, 𝑐, and 𝑑 are proportional, then 𝑎𝑑 is equal to 𝑏𝑐. And we also saw that 𝑎 over 𝑏 is equal to 𝑐 over 𝑑 which is equal to 𝑎 plus 𝑐 over 𝑏 plus 𝑑. We also saw that if three terms 𝑎, 𝑏, and 𝑐 are in continued proportion, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐, which means that 𝑎𝑐 is equal to 𝑏 squared, noting that terms are said to be in continued proportion if the ratio between successive terms is constant.