In this video, we will learn about proportion and ratio and how to find an unknown quantity in a proportion. Let’s start by thinking, what is a proportion? Imagine we have a recipe for a fruit salad, which for two people calls for two apples and three oranges. Suppose we wanted to make this fruit salad for four people instead. We need to double the quantities. We’d need four apples and six oranges instead. So we can say that, here, the apples and oranges are proportional to each other. We could say that two quantities, 𝐴 and 𝐵, are directly proportional if their ratios are equal. We might often see proportion written without the word “directly.” In this case, we could assume that it is a direct proportion. We can also write the statement, their ratios are equal, in a more mathematical way.
We can say that quantities 𝐴 and 𝐵 are directly proportional if, in a given situation, the quantity of 𝐴 is 𝐴 sub one and the quantity of 𝐵 is 𝐵 sub one. And in a different situation, the quantity of 𝐴 is 𝐴 sub two and 𝐵 is 𝐵 sub two. Then 𝐴 sub one to 𝐵 sub one is equal to 𝐴 sub two to 𝐵 sub two. And 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. If we take our apples and oranges example, then two apples to three oranges could be written as the ratio two to three. And it was equal to a ratio four to six, four apples and six oranges. We can say that they’re equal because if we reduce the ratio four to six to its simplest form, we would get two to three. As a fraction, we could have written this as two-thirds equals four-sixths. And this is because our fraction four-sixths reduces to two-thirds.
Let’s now look at how we might identify a proportional relationship. The first type could be described as a ratio that compares two parts of a whole. If we look at the ratios of the blocks below, we can see that, in the first set, the ratio of orange to pink would be three to five. In the second block, the ratio of orange to pink would be six to 10. Since we can reduce the ratio six to 10 time to three to five, then these two ratios must be a proportion. Since they’re proportional, if we can reduce the ratios to their simplest form and these are equivalent. The second type of proportion we might see is when we have a ratio comparing apart and a whole.
In our first diagram, we can see that there are 10 students and four girls. We could write this as four-tenths. In the second diagram, we have two girls and five students, which we could write as two-fifths. We could say that these would be proportional if the reduced fractions would be the same. And here they would say since we know that four-tenths reduces to two-fifths. When we’re discussing proportion, we might also come across the phrase unit rate. A unit rate is a two-term ratio, which has a second term of one. So, for example, if we wanted to find the unit rate of our ratio three to five, this means we would need an equivalent ratio with a second term is one. To get from five to one, we divide by five. So we would also need to divide the first term of our ratio by five. And we could write three divided by five as three-fifths or 0.6. So our unit rate in this case would be 0.6 to one. Let’s now have a look at some questions where we’re finding proportions.
Olivia wants to enlarge a photo that is four inches by six inches. Which of the following is proportional to the original photo. Option A) eight inches by 10 inches. Option B) 18 inches by 24 inches. Option C) 20 inches by 24 inches. Option D) 16 inches by 20 inches. Or option E) 24 inches by 36 inches.
We can start this proportion question by recalling that two quantities are proportional if their ratios are equivalent. Here, we have our photo that is four inches by six inches. We could write this as the ratio four to six. We could enlarge the photo by multiplying both the length and the width by two giving us a frame of eight inches by 12 inches. And the ratio in this case would be eight to 12. We could say that these two are in proportion since they both reduce to the ratio two to three. So let’s have a look at option A, eight inches by 10 inches. We could write this as the ratio eight to 10. However, we can immediately reject this one, since we’ve already established that our ratio would be eight to 12 and not eight to 10.
Alternatively, we could have also considered reducing the ratio eight to 10 by dividing both sides of the ratio by two giving us four to five, which is not equivalent to the ratio two to three. If you look at option B with the ratio 18 to 24, we’ll see if we could reduce this to the ratio two to three. Dividing both sides by six, we would have 18 divided by six is three and 24 divided by six is four. Since this is not equivalent to the ratio two to three, then we can reject option B.
In option C, the ratio 20 to 24 reduces to five to six by dividing both sides by four. So this can also be rejected. In option D, the ratio 16 to 20 reduces to four to five, so we can reject this. The ratio in option E, 24 to 36, reduces to two to three. And since this is the same ratio that we get by reducing our quantity of four inches by six inches, the ratio four to six, then this means that it is in proportion. So the photo that’s proportional to four inches by six inches is option E, 24 inches by 36 inches.
Charlotte can type 75 words in three minutes. Determine how many words she can type in four minutes.
We can answer this question using two different methods. The first method involves the unit rate. The unit rate is a two-term ratio, which has a second term of one. So let’s start by writing the values of 75 words and three minutes in a ratio of words to minutes. Since we’re told it’s 75 words in just three minutes, we can write this as the ratio 75 to three. To find the unit rate, we need to write this as a ratio where the second term is one. Since we can do this by dividing by three, then we also need to divide our 75 by three as well, which will give us 25. At this point, we’ve established that in one minute Charlotte would type 25 words. We now need to work out how many words she can type in four minutes. So looking at our ratio, we think, how do we go from one to four? And we would multiply by four. So we must also multiply our 25 by four, giving us the value of 100. So that’s 100 words in four minutes.
Let’s now consider an alternative method. We can say that 𝐴 and 𝐵 are directly proportional, if 𝐴 sub one to 𝐵 sub one equals 𝐴 sub two to 𝐵 sub two. And 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. Our subscripts of one and two here refer to the values of 𝐴 and 𝐵 in two different situations. So if we want to write our 75 words in three minutes as a fraction, we could write this simply as 75 over three. We can put this equal to 𝑥 over four, where 𝑥 refers to the number of words. We can then solve for 𝑥 by cross multiplying. We’re taking the cross product. We can start by writing 75 times four equals three times 𝑥. We can evaluate 75 times four as 300. And three times 𝑥 can be written as three 𝑥. To find 𝑥 by itself then, we must divide both sides of our equation by three giving us 100 equals 𝑥. So now, we know that our unknown 𝑥, the number of words, is 100 words. Using either of these methods then, we’ve established that Charlotte would type 100 words in four minutes.
Let’s take a look at another example where we can use a fraction to help us solve the proportion.
An area of land is shared between two people in the ratio 13 to 10. The first person share is 81 square metres larger than the second person’s. What is the total area of land?
In this question, we have a proportional relationship between the first person share and the second person share. When we have a proportion relationship between two values 𝐴 and 𝐵, we can say that 𝐴 sub one to 𝐵 sub one equals 𝐴 sub two to 𝐵 sub two. And 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. The values of 𝐴 sub one and 𝐵 sub one refer to the quantities of 𝐴 and 𝐵 in the first situation. And 𝐴 sub two and 𝐵 sub two refer to the values of the quantity in a second situation. So let’s start by taking our ratio of 13 to 10 and writing it in a fractional form like we can see in the definition which would be 13 over 10.
For the second equivalent fraction, we need a way to note that the first person share is 81 square metres larger than the second person’s. So let’s use a value 𝑥 to represent the size of the second person share of land. When the first person share is 81 square metres larger, we could write this as 𝑥 plus 81. So now, we need to solve this equation to find 𝑥. We can begin by taking the cross product giving us 10 times 𝑥 plus 81 equals 13 times 𝑥.
We then multiply the 10 by each term in the parenthesis, starting with 10 times 𝑥, which is 10 𝑥, and then 10 times 81, which is 810. And we can write our 13 times 𝑥 as 13 𝑥. We can continue then by subtracting 10𝑥 from both sides of our equation, giving us 810 equal three 𝑥, since 13𝑥 take away 10𝑥 is three 𝑥. To find 𝑥, we divide both sides of our equation by three. So 𝑥 equals 270 square metres. So now, we know that the second person share, the value 𝑥, is 270 square metres. To find the first person share, we would work out 270 plus 81, which is 351 square metres. We now need to work out the total area of land. So we add 351 and 270 giving us 621 square metres.
In the next question, we’ll see an example of a different kind of proportion question. We’ll need to be very careful to apply good reasoning as well as good mathematical skills.
It took three people two and one-third hours to paint a large room. How long would it take six people to paint the same room assuming that they all work at the same rate?
Let’s start by considering this as a ratio of the people to the time taken. So for three people, taking two and third hours to paint a room, we could write this as the ratio three to two and a third. We need to work out for six people how long will it take to paint the same room. It would be very easy to think we must simply multiply by two. However, we need to think carefully about the problem that’s presented. If we were painting a room and we asked someone to help us, would we expect it to take a longer time or a shorter time? We’d would expect that more people would mean that the time taken would be less. Here, we have an example of an indirect proportion. In an indirect proportion, as one variable increases, the other variable decreases. Here, as the number of people increases, the time taken would actually decrease.
So to answer this question, let’s take the variable 𝑡 to represent the time taken. We can write that three times two and a third equals six times 𝑡. The left-hand side of our equation represents the total time taken by our three people spending two and a third hours each painting the room. We can simplify our equation by writing three times seven over three equals six 𝑡, since our mixed member fraction two and a third becomes seven over three as an improper fraction. Since our threes on the left-hand side will cancel, we then have seven equals six 𝑡. To find 𝑡 then, we can divide both sides of our equation by six giving us seven over six. We could also write this as one and one-sixth. So our final answer then is that it takes six people one and one-sixth hours to paint the room.
Let’s check our answer by considering this indirect proportion. We know that it takes three people two and a third hours to paint the room. We know that we’re considering six people and we must multiply three by two to get six. The time taken by three people is two and a third hours. And we’ve established that we don’t multiply this value by two. However, since there is an indirect proportion, the values are still proportional in some way. The inverse of multiplying by two would be dividing by two, which is equivalent to multiplying by a half. So when we divide two and a third by two, we would get one and one-sixth hours, which would confirm our original answer of one and a sixth hours.
Now, let’s summarise the key points we’ve learned in this video. Quantities 𝐴 and 𝐵 are directly proportional if, in a given situation, the quantity of 𝐴 is 𝐴 sub one and 𝐵 is 𝐵 sub one. And in a different situation, the quantity of 𝐴 is 𝐴 sub two and of 𝐵 is 𝐵 sub two. Then 𝐴 sub one to 𝐵 sub one equals 𝐴 sub two to 𝐵 sub two. and 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. We could summarise this by saying two quantities are directly proportional if their ratios are equivalent.
We also learned about the unit rate, which is a two-term ratio where the second term is one. We also saw an example of a question involving indirect proportion, which is when one variable increases, the other variable decreases. So when we’re answering a question on proportion, we need to be careful to use our logic and reasoning skills as well. Since in a question involving a relationship with indirect proportion then the rules of direct proportion would not apply to them.