Question

A store sells yellow sweatshirts for $13.50 and white sweatshirts for $9.95. In one day, 28 sweatshirts are sold for a total of $349.60. How many of each colour were sold?

Answer 1

A total of 8 white and 20 yellow sweatshirts were sold.

Explanation:

Let `w` be the number of white sweatshirts sold.
Let `y` be the number of yellow sweatshirts sold.

We are given that the total number of sweatshirts sold is 28. This means:

`w+y=28`

We are also given the total sales as $349.60. This means:

`[(9.95),("$/white")][(w),("whites")]+[(13.50),("$/yellow")][(y),("yellows")]=[(349.60),($)]`

or

`9.95w + 13.5y = 349.6`

This gives us a system of two linear equations, which can be solved using many methods, including substitution:

`color(white)(=>)w+y=28`
`=> color(white)(w+)y=color(blue)(28-w)"    "` (Sub into the other equation)

`color(white)(=>)9.95w + 13.5y "             "= 349.6`
`=>9.95w+13.5color(blue)((28-w))=349.6`
`=>9.95w+378 - 13.5w" "=349.6`
`=>"                        "–3.55w=–28.4`

`=>"                                  "w=(–28.4)/(–3.55) = color(orange)8`

Now sub this back into the first equation:

`color(white)(=>)w+y=28`
`=>" "color(orange)8+y=28`
`=>"        "y=20`

So a total of 8 white and 20 yellow sweatshirts were sold.

Answer 2

`color(blue)("Method 2 of 2:")`

Modelling using the context of a graph's gradient.

20 yellow and 8 white

Explanation:

The total count of sweatshirts is fixed at 28. So if we only count the yellow ones ($13.50) then the count of white ones are directly related by: 28 - count of yellow. Consequently we can solve this by only considering the yellow. Believe it or not this can and may be represented by a straight line graph. Bear with me on this.

Suppose all 28 sweatshirts were the white ones.
For this condition the total cost would be:

`28xx$9.95 = $278.60`

Suppose all 28 sweatshirts were the yellow ones.
For this condition the total cost would be:

`28xx$13.50 = $ 378`

The gradient of part is the same as the gradient of all of it.

`("change in total cost")/("change in count of yellow") -> (378-278.6)/(28-0) = (349.6-278.6)/(x-0)`

Really you are using ratios

`99.4/28=71/x`

Turn everything upside down

`28/99.4=x/71`

`x=71xx28/99.4 = 20` yellow sweatshirts
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check:

`[20xx$13.50]+[8xx$9.95] = $ 349.60`

Answer 3

`color(blue)("Method 1 of 2")`

20 yellow sweatshirts and 8 white.

Explanation:

Let the count of yellow sweatshirts be `x`

Then the count of white is `color(white)("b")28-x color(white)("b")larr" As in my graph model"`

Thus we have:

(Count yellow x cost yellow) + (count white x cost white) = $349.60

`color(white)("bbb")( x xx $13.50)color(white)("mmmmb")+color(white)("mmb")$9.95(28-x)color(white)("mb")=$349.6`

Dropping the $ sign

`13.5x+278.6-9.95x=349.6`

`3.55x=71`

`x=20` yellow sweatshirts