Question

You need to spend exactly $20 to buy eggs at $1 per carton of a dozen eggs and cheese at $2 per block containingeight ounces of cheese to make omelettes There must be exactly twice as many eggs as ounces of cheese in theomelettes. How many cartons of eggs and blocks of cheese should you buy?Next think about what equation represents the fact that there must be twice as many eggs as ounces of cheese inthe omelette. Letx=the number of cartons of eggs (with 12 eggs in each carton) andy=the number of blocks of cheese (with 8 ounces of cheese in each block).

Answer 1

1. i. The number of cartons of eggs to buy is 10

ii. The number of blocks of cheese to buy is 5

2. The equation representing the statement "twice as many eggs as ounces of cheese" is:

`X=2Y`

SOLUTION

Problem Statement

The question says that we need to spend exactly $20 and we are required to buy exactly 2 times as many cartons of eggs as blocks of cheese.

We are asked to:

1. Find the number of cartons of eggs and blocks of cheese to buy.

2. The equation representing the statement "twice as many eggs as ounces of cheese"

Solution

To solve this problem, we need to interpret certain statements within the question and re-write them as equations. We shall get two equations and solve them simultaneously and their solution would give us the number of cartons of eggs and blocks of cheese to buy.

Step 1:

The first statement is:

"You need to spend exactly $20 to buy eggs at $1 per carton of a dozen eggs and cheese at $2 per block containing

eight ounces of cheese"

This means that the cost of eggs added to the cost of the blocks of cheese must add up to $20.

If 1 carton of eggs costs $1 and we buy X cartons of eggs, it means that:

`\begin{gathered} \text{The cost of X cartons of eggs is:} \\ X*\text{ \$1=\$X} \end{gathered}`

If 1 block of cheese is $2, and we buy Y blocks of cheese, it means that:

`\begin{gathered} \text{The cost of Y blocks of cheese is:} \\ Y*\text{ \$2= \$2Y} \end{gathered}`

Now that we know the cost of all the eggs we want to buy ($ X) and the cost of all the cheese blocks ($Y), we can add them up and equate them to the total cost, $20.

`X+2Y=20\text{ (Equation 1)}`

Step 2

The next statement we need to interpret is that part of the question that says:

"There must be exactly twice as many eggs as ounces of cheese in the omelettes"

This means that when we count the cartons of eggs, they must outnumber the blocks of cheese we have by a factor of two.

For example,

if we bought 3 blocks of cheese, then, we must have bought 6 (i.e. 3 x 2) cartons of eggs or

if we bought 6 blocks of cheese, it means we must have bought 12 (i.e. 6 x 2) cartons of eggs.

Thus, if X represents cartons of eggs and Y represents blocks of cheese, it means that

if we bought Y blocks of cheese, then, we must have bought X (i.e. Y x 2) cartons of eggs.

Therefore, the second equation can be written as:

`X=2Y\text{ (Equation 2)}`

Step 3

Now that we have the two equations, we can solve them simultaneously.

Equation 2 makes X the subject of 2Y, thus, we can substitute this expression for X into Equation 1 to find the value of Y (i.e. the number of blocks of cheese to buy)

`\begin{gathered} X=2Y \\ \text{Substituting this expression for X into Equation 1, we have:} \\ X+2Y=20\text{ (Equation 1)} \\ 2Y+2Y=20 \\ 4Y=20 \\ \text{Divide both sides by 4} \\ (4Y)/(4)=(20)/(4) \\ \therefore Y=5 \end{gathered}`

Therefore, we must buy 5 blocks of cheese.

Next, let us find the number of egg cartons to buy. This can be done by simply substituting the value of Y into any of equations 1 or 2.

Substituting into equation 2, we have:

`\begin{gathered} X=2Y \\ \text{But Y = 5} \\ X=2*5 \\ \\ \therefore X=10 \end{gathered}`

Therefore, we must buy 10 cartons of eggs

Final Answer

1. i. The number of cartons of eggs to buy is 10

ii. The number of blocks of cheese to buy is 5

2. The equation representing the statement "twice as many eggs as ounces of cheese" is:

`X=2Y`