In this video, we will learn how to use our understanding of ratios to solve word problems. We will begin by recapping what we mean by the word ratio and how it can be used in mathematics.
A ratio shows the relative sizes of two or more values. For example, if we had 12 boys and eight girls in a class, then the ratio of boys to girls can be written as 12 to eight, where we use a colon to separate the values. This ratio can be simplified by dividing both sides by four. 12 divided by four is equal to three. And eight divided by four is equal to two. This means that the ratio of boys to girls, in its simplest form, is three to two. This simplified ratio can be used to find the fraction of the class that are boys and the fraction that are girls. Three out of the five parts are boys. Therefore, three-fifths of the class are boys. As two parts represented the number of girls, then two-fifths of the class are girls.
In this video, we will use ratios to solve problems involving scaling, equivalent ratios, sharing a quantity, and recipe problems.
If seven books cost 14 dollars, use a ratio table to determine the cost of eight books.
In this question, we need to consider the number of books and the cost. We will assume that each book cost the same value. We are told in the question that seven books cost 14 dollars. We are asked to calculate the cost of eight books. We can do this by finding the unit cost first. In this case, this is the cost of one book. The unit cost is equal to the total cost divided by the number of units.
In this question, we can calculate the cost per book by dividing 14 by seven. This is equal to two. Therefore, each book cost two dollars. We have divided the number of books and the cost by seven. We can now calculate the cost of eight books by multiplying two dollars by eight. Two multiplied by eight is equal to 16. Therefore, the cost of eight books is 16 dollars.
Finding the unit cost first allows us to solve problems like this using scaling. We can calculate the total cost of any number of books by multiplying the unit cost by the number of books.
Our next question is a problem involving equivalent ratios.
Charlotte wants to enlarge one of her photos. The original size of the photo is three by five inches. Given that the enlarged photo will have a height of 10 inches, determine the width of the enlarged photo.
We are told in the question that the original photo is three inches by five inches. This means that the ratio of the width to the height is three to five. The height of the enlarged photo is 10 inches. And we need to calculate its width. If we let this width equal 𝑥 inches, then the ratio of width to height is 𝑥 to 10. As five multiplied by two is equal to 10, we can see that the scale factor of enlargement is two.
As we have multiplied the height by two, we also need to multiply the width by two. Whatever we do to one side of the ratio we must do to the other. Three multiplied by two is equal to six. This means that the width of the enlarged photo is six inches. The ratio three to five is equivalent to the ratio six to 10.
The next question we will look at involves finding the values of two quantities given their sum and the ratio between them.
For school uniforms at the beginning of the school year, a mother shares 370 Egyptian pounds between her sons William and Daniel in the ratio seven to three. How much money did each son receive?
In this question, we are given the total amount of money, 370 Egyptian pounds, and the ratio this is shared, seven to three. For every seven pound William gets, Daniel gets three pound. The order of the ratio is important here.
In order to share any value in a given ratio, we need to use the following steps. Our first step is to add the ratios. In this case, seven plus three is equal to 10. Our next step is to divide the total by this answer. This enables us to calculate the value of one share or one part. 370 divided by 10 is equal to 37. Therefore, each part of the ratio is worth 37 Egyptian pounds. Our final step is to multiply each part of the ratio by this value.
We need to multiply seven by 37 and three by 37. Seven multiplied by 37 is 259 as seven multiplied by 30 is 210 and seven multiplied by seven is 49. Adding these two numbers gives us 259. Three multiplied by 37 is 111. At this stage, it is worth checking that the sum of our answers is 370. 259 plus 111 is indeed 370. As William gets the larger share, we can conclude that he gets 259 Egyptian pounds and Daniel gets 111 Egyptian pounds.
The penultimate question in this video involves writing and solving a system of linear equations in order to solve a problem involving ratios.
Given that the perimeter of a rectangle equals 72 centimeters and the ratio between the lengths of two of its sides is five to four, determine its area.
Let’s consider the rectangle as shown with length 𝑥 centimeters and width 𝑦 centimeters. We are told that its perimeter is equal to 72 centimeters. And the perimeter of a rectangle is the distance around the outside. As the opposite sides of a rectangle are parallel and equal in length, we have the equation 𝑥 plus 𝑦 plus 𝑥 plus 𝑦 is equal to 72. Grouping or collecting like terms, this equation simplifies to two 𝑥 plus two 𝑦 is equal to 72.
We can divide both sides of the equation by two, giving us 𝑥 plus 𝑦 is equal to 36. We are also told in the question that the ratio between the lengths of the two sides is five to four. Therefore, 𝑥 and 𝑦 are in the ratio five to four. We now know the sum of the two quantities and the ratio between them.
We can calculate one part of the ratio by dividing the sum of the quantities by the number of parts. In this case, one part or one share is equal to 36 divided by nine. This is equal to four. Our next step is to multiply each of the parts by four. Five multiplied by four is equal to 20. And four multiplied by four is equal to 16. This means that the lengths 𝑥 and 𝑦 are 20 centimeters and 16 centimeters, respectively.
We were asked to determine the area of the rectangle. We do this by multiplying the length by the width. Two multiplied by 16 is equal to 32. Therefore, 20 multiplied by 16 is 320. The area of the rectangle is 320 square centimeters.
The final question we will look at in this video involves recipes.
The ratio of the number of cups of diced tomatoes to the number of cups of diced cucumbers for a salad recipe is nine to seven. If a salad recipe called for three-sevenths of a cup of diced tomatoes, determine how many cups of diced cucumbers are needed.
We are told in the question that the ratio of tomatoes to cucumbers is nine to seven. We need to work out the number of cups of diced cucumbers that would be required for a recipe involving three-sevenths of a cup of diced tomatoes. There are lots of ways of approaching this problem. But remember, whatever we do to one side of the ratio we must do to the other.
Nine divided by three is equal to three. Therefore, we could work out the number of cups of diced cucumbers required for three cups of diced tomatoes by dividing seven by three. This is equal to seven-thirds. Three divided by seven is equal to three-sevenths. We could, therefore, divide seven-thirds by seven to work out the number of cups of diced cucumbers required. Seven-thirds divided by seven is equal to one-third. If a salad recipe called for three-sevenths of a cup of diced tomatoes, we would need one-third of a cup of diced cucumbers.
An alternative method which is often used in recipe problems would be to find the number of cups of cucumbers required for one cup of tomatoes first. This would work out the unit value and would allow us to scale up and down as required.
We will now summarize the key points from this video.
Ratios are used to show the relative sizes of two or more values. For example, the ratio one to three means that for every one unit of the first part, we have three units of the second part. These can be converted to fractions by writing each part as a fraction of the whole. The one represents one-quarter of the total and the three represents three-quarters. This could also be written in decimal or percentage form, 0.25 and 0.75 or 25 percent and 75 percent.
We can solve a variety of word problems involving scaling, equivalent ratios, sharing a quantity in a given ratio, and recipe questions. The majority of these type of ratio questions involve finding the unit value first. This involves calculating what one part is equal to.