Question

Molly's Custom Kitchen Supplies sells handmade forks and spoons. It costs the store $1.70 to buy the

supplies to make a fork and $1.30 to buy the supplies to make a spoon. The store sells forks for $5.60
and spoons for $5.40. Last April Molly's Custom Kitchen Supplies spent $37.90 on materials for forks
and spoons. They sold the finished products for a total of S147.20. How many forks and how many
spoons did they make last April?
Write the equation for the cost of making forks and spoons:
Write the equation that represents the money made from selling forks and
spoons:
They made
forks "type in the number only
They made
spoons *type in the number only

Answer 1

9514 1404 393

Answer:

• 7 forks
• 20 spoons

Explanation:

Let x and y represent the numbers of forks and spoons made, respectively.

The cost of materials is ...

1.70x + 1.30y = 37.90

The revenue from sales is ...

5.60x + 5.40y = 147.20

You can solve these by any of your favorite methods. A graph is attached. It shows the solution is (x, y) = (7, 20).

They made 7 forks and 20 spoons.

_____

Additional working out

The equation coefficients are not "nice" multiples of each other, so elimination and substitution will require multiplying both equations by some numbers. If we're going to that trouble, we may as well use Cramer's Rule or a variation of the "cross multiplication method". The latter is relatively straightforward and can be easy to remember. It tells you the solution to ...

ax +by +c = 0

dx +ey +g = 0

is ...

1/(ae-db) = x/(bg-ey) = y/(cd-ga)

Here, that becomes ...

1/(1.70·5.40-5.60·1.30) = x/(1.30·(-147.20)-5.40·(-37.90))

= y/(-37.90·5.60-(-147.20·1.70))

1/1.9 = x/13.3 = y/38

x = 13.3/1.9 = 7 . . . number of forks

y = 38/1.9 = 20 . . . number of spoons

__

The above solution formula may be more easily remembered in terms of the cross-multiplication pattern that gives the method its name. When the coefficients are written in two rows of four, with the last being a repeat of the first:

• a b c a
• d e g d

each of the three denominators (of 1, x, and y) is the difference of products in adjacent columns, formed in an X fashion: ae-db, bg-ec, cd-ga. Note that the equations are written in "general" form, which is similar to "standard" form, but with all of the non-zero terms on the left of the equal sign. (This has the effect of changing the sign of the constants: 1.7x+1.3y=37.9 ⇒ 1.7x+1.3y-37.9=0)