### Video Transcript

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 π₯ inches,
then the ratio of width to height is π₯ 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.

Letβs consider the rectangle as shown
with length π₯ centimeters and width π¦ 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 π₯ plus π¦ plus π₯ plus π¦ is equal
to 72. Grouping or collecting like terms, this
equation simplifies to two π₯ plus two π¦ is equal to 72.

We can divide both sides of the
equation by two, giving us π₯ plus π¦ 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, π₯ and π¦ 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 π₯ and π¦
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.