Video Transcript
In this video, we will look at
comparing ratios using unit rates in real-life problems. We will begin by recapping how we
can simplify ratios. If we consider the ratios four to
10 and six to 18, we can simplify them by looking for common factors. Four and 10 have a highest common
factor of two, so we can divide both sides of the ratio by two. This tells us that the ratio four
to 10 in its simplest form is two to five.
We can repeat this process for the
ratio six to 18. The highest common factor this time
is six. As six divided by six is equal to
one and 18 divided by six is equal to three, the ratio in its simplest form is one
to three. This is all well and good if we
want a ratio in its simplest form. But what if we wanted to compare
the two ratios? How can we compare two to five and
one to three? In order to compare two or more
ratios, we need to write them in the form one to 𝑛. This is known as the unit ratio or
unit rate.
In terms of our example, the second
ratio, one to three, is already written in this form. This means that, for every one unit
of the first part, we get three units of the second part. In order to write the ratio two to
five as a unit ratio, we need to divide both sides by two. Two divided by two is equal to
one. Five divided by two can be written
as the fraction five-halves or five over two. When comparing ratios, it is useful
to turn this fraction into a decimal. The ratio four to 10 or two to five
written as a unit ratio is one to 2.5. For every one unit of the first
part, we get two and a half or 2.5 units of the second part. We’re now in a position to compare
the ratios as required in the question.
An alternative method to compare
ratios is to consider the first part as a fraction of the whole. In our ratio four to 10, the first
part four is four out of 14 parts altogether. In the same way, the first part of
the second ratio, six to 18, is six parts out of 24 in total. Both of these fractions can be
simplified by dividing the numerator and denominator by two and six,
respectively.
We still have a problem when trying
to compare two-sevenths and one-quarter. The easiest way to do so would be
to find the lowest common multiple of four and seven. This is 28. Two-sevenths is equivalent to eight
twenty-eighths, whereas one-quarter is equivalent to seven twenty-eighths. As the denominators are now equal,
we can compare the fractions by looking at the numerators. Whilst the second method is
sometimes useful, for the majority of this video, we will compare ratios using unit
ratios or unit rates.
Scarlett uses two tablespoons
of sugar for every three glasses of lemonade, while Natalie uses three
tablespoons of sugar for every six glasses of lemonade. Who makes the sweeter
lemonade?
We can begin this question by
writing down a ratio of sugar to lemonade for both of the girls. Scarlett used two tablespoons
for every three glasses of lemonade. Therefore, her ratio is two to
three. Natalie used three tablespoons
of sugar for every six glasses of lemonade. So, her ratio is three to
six. One way of comparing two or
more ratios is to write them in the form one to 𝑛. This is known as the unit
ratio. When finding an equivalent
ratio, we must divide or multiply both sides by the same value. Two divided by two is equal to
one. And three divided by two is
equal to 1.5. This means that, for every one
tablespoon of sugar that Scarlett uses, she will fill 1.5 glasses of
lemonade.
Repeating this process for
Natalie, we divide both sides of her ratio by three. The ratio three to six
simplifies to one to two. This means that, for every one
tablespoon of sugar that Natalie uses, she is able to fill two glasses of
lemonade. We’re asked to work out who
makes the sweeter lemonade. This will be the person who has
less glasses per tablespoon of sugar. As Scarlett is only able to
make one and a half glasses of lemonade for one tablespoon of sugar, she makes
the sweeter lemonade.
Had we noticed that Natalie had
double the number of glasses originally than Scarlett, we could’ve used an
alternative method for this question. Multiplying both sides of
Scarlett’s ratio by two gives us a new ratio, four to six. As Scarlett used four
tablespoons of sugar for six glasses of lemonade whereas Natalie only used three
tablespoons, once again, we have proven that Scarlett made the sweeter
lemonade.
Michael wants to sign up for
his school’s sports competition. In order to be accepted, he has
to be able to run 400 meters in one minute. Michael took 20 seconds to run
100 meters. If he could run at the same
rate, would he qualify to take part?
We’re told in the question that
Michael needs to be able to run 400 meters in one minute. If we write this as a ratio of
time in minutes to distance in meters, this would be one to 400. We’re also told that Michael
can run 100 meters in 20 seconds. There are 60 seconds in one
minute. And 20 multiplied by three is
equal to 60. Multiplying 100 by three gives
us 300. So, if Michael runs at the same
rate, he will cover 300 meters in 60 seconds. This ratio of time in minutes
to distance in meters is one to 300.
As Michael needed to cover a
distance of 400 meters in one minute, the correct answer is no. He would not qualify to take
part. Writing any ratio in this form,
one to 𝑛, is known as the unit ratio or unit rate. For every one unit, or minute
in this case, of time, we can see the distance in meters that Michael
covered.
The next question we look at, we’ll
look at unit rates to compare three different ratios.
Mason, Liam, and James are
biking. Mason can bike two miles in 20
minutes, Liam can bike three miles in 25 minutes, and James can bike six miles
in 66 minutes. Who cycles at the fastest
rate?
In order to compare the three
speeds, we will write the ratio of distance in miles to time in minutes. For Mason, this is a ratio of
two to 20. For Liam, the ratio is three to
25. And finally, for James, the
ratio is six to 66. In order to compare the three
ratios, we need to calculate the unit rate or unit ratio. This is written in the form one
to 𝑛. In this question, this will
calculate the time it would take each boy to cycle a distance of one mile. When simplifying or finding
equivalent ratios, we need to multiply or divide both sides by the same
number. For mason, we need to divide
both sides by two. This means that the ratio two
to 20 is equivalent to one to 10. It takes Mason 10 minutes to
bike one mile.
To make the left-hand side of
Liam’s ratio equal to one, we need to divide both sides by three. 25 divided by three is equal to
eight and one-third or 8.3 recurring, written with a dot or bar above the
three. It takes Liam eight and a third
minutes to cycle one mile. Dividing both sides of James’s
ratio by six gives us the new ratio one to 11. It takes James 11 minutes to
cycle one mile. As all three ratios are now
written in terms of their unit rate, we can compare them. The person who cycles at the
fastest rate will be the person who takes the least amount of time to cycle one
mile. In this question, this is
Liam.
The next question involves
comparing ratios using tables.
Use the table to determine
which runners ran at the same rate.
In order to answer this
question, we will look at the ratio of each runner of their time in hours to
distance in miles. For Liam, this is two to
10. He jogged 10 miles in two
hours. James’s ratio was three to 18
as he ran 18 miles in three hours. The corresponding ratios for
David and Michael were four to 20 and three to 12, respectively. In order to compare two or more
ratios, we need to write them in the form one to 𝑛. This is the unit rate or unit
ratio. In this question, it will be
the distance that each runner jogged in one hour.
For Liam, we will divide both
sides of the ratio by two. This means that Liam ran a
distance of five miles per hour. We divide the two parts of
James’s ratio by three. This gives us a ratio of one to
six. So, James ran six miles in one
hour. Repeating this for David and
Michael tells us that David ran five miles per hour and Michael ran four miles
per hour. We’ve been asked to identify
which runners ran at the same rate. As both Liam and David had the
same unit ratio of one to five, we can conclude that they ran at the same
rate.
Our final question will involve
using a calculator to calculate density to compare populations.
Daniel and Charlotte both have
a keen interest in gardening and are concerned by the number of slugs that they
keep finding in their respective vegetable patches. They want to compare the number
of slugs in each of their gardens. But due to the differences in
size of their vegetable patches, they decide to compare the number of slugs per
square foot. Daniel’s vegetable patch is a
rectangle with dimensions five foot by three foot. And Charlotte’s is a circular
patch with a radius of three foot. One Saturday morning, Daniel
counts 21 slugs in his entire vegetable patch and Charlotte counts 36. There are three parts to this
question. Work out the density of slugs
in Daniel’s vegetable patch. Work out the density of slugs
in Charlotte’s vegetable patch. Who has the more severe slug
problem?
We’re told that Daniel’s patch
is rectangular and measures five foot by three foot, whereas Charlotte’s is
circular with a radius of three foot. There were 21 slugs in Daniel’s
vegetable patch and 36 in Charlotte’s. We will now clear some space to
calculate the number of slugs they had per square foot. Let’s consider Daniel’s
vegetable patch first. His patch was rectangular with
dimensions three foot and five foot, and he found 21 slugs in his patch. We can calculate the area of
any rectangle by multiplying the length by the width. In this case, we need to
multiply five by three. This is equal to 15. Therefore, Daniel’s patch has
an area of 15 square foot.
In order to calculate the
density per square foot, we can, firstly, write the ratio of the area to the
number of slugs. This is equal to 15 to 21. To calculate the density of
slugs in Daniel’s patch, we need to calculate the unit ratio, how many slugs
there are per square foot. This is written in the form one
to 𝑛. We divide both sides of the
ratio by 15, giving us the ratio one to 1.4. The density of slugs in
Daniel’s vegetable patch is therefore 1.4 slugs per square foot.
We can now repeat this process
for Charlotte. Charlotte’s vegetable patch was
circular and had a ratio [radius] of three foot. She found 36 slugs in her
vegetable patch. The area of any circle can be
calculated by multiplying 𝜋 by the radius squared. In this question, this is equal
to 𝜋 multiplied by three squared. This is equal to 28.2743 and so
on. This means that the area of
Charlotte’s vegetable patch is 28.27 square feet. For the purposes of this
question, we will keep this as nine 𝜋. The ratio of area to slugs for
Charlotte is therefore nine 𝜋 to 36. To find the unit ratio or
density, we can divide both sides by nine 𝜋. 36 divided by nine 𝜋 is equal
to 1.273 and so on. Rounding this to one decimal
place gives us 1.3 slugs per square foot.
The three correct answers are
1.4, 1.3, and Daniel. As 1.4 is greater than 1.3,
Daniel has the more severe slug problem.
We will finish this video by
summarizing the key points. In order to compare two or more
ratios, we need to calculate the unit rate or unit ratio. This is written in the form one to
𝑛. To simplify a ratio or find an
equivalent ratio, we must multiply or divide all parts of the ratio by the same
number. For example, the ratio four to 12
can be written as a unit ratio by dividing both parts by four. The ratio four to 12 is equivalent
to the unit ratio one to three. For every one unit of the first
part, we have three units of the second part.
