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Grade 9

2.02 Applying ratios

Lesson

Proportion

When $2$2 rates or ratio are equal (equivalent) they are in proportion. The following ratios are in proportion:

$3:5=$3:5=$15:25=$15:25=$\frac{1}{5}:\frac{1}{3}$15:13

You can use this concept of proportion to find a missing value by making equivalent ratios. If a recipe calls for $2$2 eggs for every $3$3 cups of flour, how many eggs are needed for $15$15 cups of flour? This can be written as:

$2:3$2:3 $=$= $x:15$x:15

Ratios are equal (in proportion)

$\frac{2}{3}$23 $=$= $\frac{x}{15}$x15

Expressing them as fractions

$\frac{2}{3}\times15$23×15 $=$= $x$x

Solving for $x$x

$x$x $=$= $10$10

Simplifying

 

You would need $10$10 eggs.

Alternatively, you could use equivalent fractions and make the denominators the same. In this case, multiplying the top and bottom of the left-hand-side fraction by $5$5 would give the same answer.

Practice question

Question 1

The two quantities are in proportion. Find the missing value.

  1. $\frac{\editable{}}{10}:\frac{35}{50}$10:3550

 

Dividing quantities using ratio

You may wish to divide a quantity by a given ratio.

If we were dividing a quantity of $4$4 items using a ratio of $1:3$1:3, it could be represented using these blue and green dots.

Here there are $4$4 parts in the ratio and the quantity being divided is $4$4. What happens if we have $40$40 items and we want to divide them in the ratio $1:3$1:3?

First, calculate the total number of parts in the ratio, then use it to divide the quantity into a given ratio.

The total number of parts in the (part-part) ratio is found by adding all the parts. In this case $1+3=4$1+3=4. Then we can divide the total quantity, which is $40$40 in this case by the total number of parts, which is $4$4, to give $10$10. Then using the ratio, you have a blue group of $1\times10=10$1×10=10 and a green group of $3\times10=30$3×10=30. Here we've multiple each term in the ratio by $10$10.

To divide a quantity by a ratio you first identify the number of parts in the ratio.

A ratio of $3:8:1$3:8:1 would have $12$12 parts in total.

Practice questions

Question 2

Find the total number of parts in the following ratios:

  1. $2:3$2:3

  2. $7:17$7:17

  3. $73:53$73:53

  4. $5:7:9$5:7:9

Question 3

$25.9$25.9 is divided into two parts, $A$A and $B$B, in the ratio $5:2$5:2.

  1. What is the value of $A$A?

  2. What is the value of $B$B?

Question 4

The perimeter of a rectangle is $110$110 cm and the ratio of its length to its width is $6:5$6:5.

  1. How many parts are in the ratio?

  2. What is the sum of the length and width of the rectangle?

  3. What is the length of the rectangle?

  4. What is the width of the rectangle?

  5. What is the area of the rectangle?

  6. What is the ratio of the area to the perimeter?

 

The unitary method

This is a method of carrying out a calculation to find the value of a number of items by first finding the cost of one of them. This method of solving problems is often handy for solving word problems.

Practice questions

Question 5

Buzz bought $6$6 stamps for $24.

  1. What is the price for 1 stamp?

  2. How much would it cost him if he only wants to buy $2$2 stamps?

Question 6

$3$3 workers can build $5$5 tables in $8$8 minutes.

  1. How long would it take for 1 worker to build $5$5 tables?

  2. How long would it take for $2$2 workers to build the same number of tables ?

 

Convert units

Another application of ratios can also be seen in converting units, imperial to metric and vice versa.

Worked Examples

example 1

The ratio of miles to kilometers is $1:1.6$1:1.6. Use this fact to find out how many kilometers are equal to $5$5 miles.

Think:  The given ratio of $1:1.6$1:1.6 means that $1$1 mile is equal to $1.6$1.6 kilometers.  Set up a ratio problem, identifying by what factor each part of the ratio needs to be multiplied to give $5$5 miles.

Do

miles to  kilometers
$1$1 $:$: $1.6$1.6
to get $5$5 miles, we need $5$5 lots of  the part representing miles in the ratio   so we also need to multiply the number of kilometers by $5$5
$1\times5$1×5 $:$: $1.6\times5$1.6×5
$5$5 $:$: $8$8

So $5$5 miles is equivalent to $8$8 kilometers.

 

example 2

$1$1 gallon is around $3.8$3.8 liters.

a)  State this as a ratio of liters to gallons, in the form $a:b$a:b. Give your answer in simplest form.

Think: We want a ratio that compares the number of liters to the numer of gallons. 

Do: Litres:Gallons = $3.8:1$3.8:1

Simplest form is without fractions or decimals:

$3.8$3.8 $:$: $1$1 multiply both parts by $10$10 to remove the decimal
$38$38 $:$: $10$10 divide both parts by common factor of $2$2
$19$19 $:$: $5$5 this is in simplest form

This means that $19$19 litres is equivalent to $5$5 gallons.

 

b) How many liters would a $15$15 gallon vat hold?

Think: Use the conversion ratio.

Do

Litres to Gallons  
$19$19 $:$: $5$5 for $15$15 gallons we need to multiply the $5$5 "gallons" parts by $3$3.
$19\times3$19×3 $:$: $5\times3$5×3 so we need to multiply both parts of the ratio by $3$3
$57$57 $:$: $15$15  

So $15$15 Gallons is equivalent to $57$57 Litres. 

 

Practice Questions

Question 7

The ratio of kilograms to pounds is $1:2.2$1:2.2. Use this fact to complete the workings below for finding out how many pounds are equal to $10$10 kilograms.

  1.     $1$1 : $2.2$2.2    
    × $\editable{}$       × $10$10
        $\editable{}$ : $\editable{}$    

 

Question 8

$1$1 foot is approximately $0.3$0.3 meters.

  1. State this as a ratio of feet to meters, in simplest form.

  2. How many meters would a $seven$seven foot garden be?

 

 

Outcomes

9.B3.3

Apply an understanding of integers to explain the effects that positive and negative signs have on the values of ratios, rates, fractions, and decimals, in various contexts.

9.B3.5

Pose and solve problems involving rates, percentages, and proportions in various contexts, including contexts connected to real-life applications of data, measurement, geometry, linear relations, and financial literacy.

9.E1.3

Solve problems involving different units within a measurement system and between measurement systems, including those from various cultures or communities, using various representations and technology, when appropriate.

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