Ratios and Rates

Ontario 09 Academic (MPM1D)

Dividing a quantity into a given ratio

Lesson

In Looking at Relationships Between Different Groups, we learnt that ratios can be used to divide up things such as quantities of money, weights and measurements. To work out the total number of parts in a ratio, we add up all the individual parts in that ratio. Look at the picture below:

In this picture, we have $1$1 blue dot to $3$3 green dots, making a total of $4$4 dots. If we wanted to write the ratio of blue to green dots, we would write $1:3$1:3. To work out the total number of parts in the ratio, we would find the sum of all the parts. In this case $1+3=4$1+3=4.

How many parts are in the ratio $22:15$22:15?

To work this out, we do $22+15$22+15. So there are $37$37 parts in total.

It doesn't matter how many parts there are to the ratio, we just keep adding them up to get the total number of parts.

What is the total number of parts in the ratio $12:3:7:11$12:3:7:11 ?

$12+3+7+11=33$12+3+7+11=33

That means the total number of parts is $33$33.

Once you can calculate the total number of parts, we can use it to divide up quantities in a given ratio.

** **$25.9$25.9 is divided up into two parts, $A$`A` and $B$`B`, in the ratio $5:2$5:2. How much is each part worth?

a) What is the value of A?

b) What is the value of B?

Neil and Dave share $\$77$$77 in the ratio $5:2$5:2.

**a)** What fraction of the total amount to be shared does Dave receive?

**b)** Therefore how much money does Dave receive?

Here's another example that combines a few different math concepts that you've learnt so far.

The perimeter of a rectangle is $198$198 and the ratio of length to width is $6:5$6:5.

**a)** Given that the length of the rectangle is $x$`x`, write an expression for the width in terms of $x$`x`.

**b)** Write an expression for the perimeter of the rectangle of $x$`x` and simplify that expression.

**c)** Hence, find $x$`x`, the length of the rectangle.

**d) **Hence find the width of the rectangle.

Solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion