The Magic of Algebra: Solving Word Problems with Equations

[mathdad]

Three apples cost as much as 4 pears. Three pears cost as much as 2 oranges. How many apples cost as much as 72 oranges?

Is this a proportion set up?

If so, can you set it up?

 

[MarkFL]

The way I would go about this one is to let P be the cost of a single pear, A be the cost of a single apple, and O be the cost of a single orange. From the information provided, we may state:

\(\displaystyle 3A=4P\)

\(\displaystyle 2O=3P\)

Now, we want to relate apples and oranges, so I would multiply the first equation by 3, and the second equation by -4, and add the equations, which will eliminate P, and result in an equation relating apples to oranges. At that point, you want 72O on one side, which will tell you how many apples are equivalent in price to 72 oranges.

 

[mathdad]

The way I would go about this one is to let P be the cost of a single pear, A be the cost of a single apple, and O be the cost of a single orange. From the information provided, we may state:

\(\displaystyle 3A=4P\)

\(\displaystyle 2O=3P\)

Now, we want to relate apples and oranges, so I would multiply the first equation by 3, and the second equation by -4, and add the equations, which will eliminate P, and result in an equation relating apples to oranges. At that point, you want 72O on one side, which will tell you how many apples are equivalent in price to 72 oranges.

I did as you suggested and got 144 apples.
I then divided 144 by 3 and 48 apples.

The choices for the number of apples are 36, 48, 64, 81.

In the back of the book, the explanation is tricky and reveals the answer to 81 apples.

Here is the explanation according to the author:

Let A = cost of 1 apple.
Let P = cost of 1 pear.
Let (Or) = cost of 1 orange.
(1) 3A = 4P So, P = (3/4)A Substitute into (2) below.

(2) 3P = 2(Or)
(2a) 3(3/4)A = 2(Or) Divide both sides by 2.
(Or) = (9/8)A Multiply both sides by 72.
72(Or) = 72(9/8)A
72(Or) = 81A

So, 81 Apples cost the same as 72 Oranges.

I was seeking an easier explanation than the book's solution.

 

[MarkFL]

I did as you suggested and got 144 apples.
I then divided 144 by 3 and 48 apples.

The choices for the number of apples are 36, 48, 64, 81.

In the back of the book, the explanation is tricky and reveals the answer to 81 apples.

Here is the explanation according to the author:

Let A = cost of 1 apple.
Let P = cost of 1 pear.
Let (Or) = cost of 1 orange.
(1) 3A = 4P So, P = (3/4)A Substitute into (2) below.

(2) 3P = 2(Or)
(2a) 3(3/4)A = 2(Or) Divide both sides by 2.
(Or) = (9/8)A Multiply both sides by 72.
72(Or) = 72(9/8)A
72(Or) = 81A

So, 81 Apples cost the same as 72 Oranges.

I was seeking an easier explanation that the book's solution.

I also got 81 apples...here's what I did:

\(\displaystyle 3A=4P\)

\(\displaystyle 2O=3P\)

Now, we want to relate apples and oranges, so I would multiply the first equation by 3, and the second equation by -4, and add the equations, which will eliminate P, and result in an equation relating apples to oranges. At that point, you want 72O on one side, which will tell you how many apples are equivalent in price to 72 oranges.

\(\displaystyle 9A=12P\)

\(\displaystyle -8O=-12P\)

Adding the equations, we get:

\(\displaystyle 9A-8O=0\)

or:

\(\displaystyle 9A=8O\)

We want 72O on one side, so multiply through by 9:

\(\displaystyle 81A=72O\)

So, we find 81 apples costs the same as 72 oranges. :D

 

[mathdad]

I also got 81 apples...here's what I did:

\(\displaystyle 3A=4P\)

\(\displaystyle 2O=3P\)

Now, we want to relate apples and oranges, so I would multiply the first equation by 3, and the second equation by -4, and add the equations, which will eliminate P, and result in an equation relating apples to oranges. At that point, you want 72O on one side, which will tell you how many apples are equivalent in price to 72 oranges.

\(\displaystyle 9A=12P\)

\(\displaystyle -8O=-12P\)

Adding the equations, we get:

\(\displaystyle 9A-8O=0\)

or:

\(\displaystyle 9A=8O\)

We want 72O on one side, so multiply through by 9:

\(\displaystyle 81A=72O\)

So, we find 81 apples costs the same as 72 oranges. :D

You just demonstrated the beauty of algebra. To take a tricky application and translate it to equations leading to the answer is the true work of a mathematician. Not too many people can reason this way.

If I could master this art of solving word problems by translating words to equations, I probably would be working for Kaplan, Sylvan, Princeston and/or other prestigious tutoring companies making tons of money. Tutoring is better than classroom teaching. Algebra is true magic.