Word Problem

Question

I bought a certain number of oranges at three for a dollar and five sixths of that number at four for a dollar. If i sold all my oranges at sixteen for six dollars, I would make a profit of three and half dollars. How many oranges did i buy?

Answer 1

We gotta make an equation out of this, so we first gotta think about whether our terms will be measuring amounts of money or amounts of oranges. Numbers of oranges is what we need to find out, but those different money rates is what helps us find the correct number of oranges. So we choose money. We will write how much money each of those different groups of oranges sold for as [how much each orange is worth]×[how many oranges were sold]. Let the "certain number of oranges" be A.

• You bought A oranges at 3 for $1.00, so each orange is worth $1/3
• So you paid out of pocket (1/3)×A dollars for those A oranges
• Next you bought (5/6)×A at 4 for $1.00, so each orange there is worth $1/4
• You paid out of pocket (1/4)×(5/6)×A = (5/24)×A dollars for those 5/6^th of that number of oranges
• Before we move on: note that the total number of oranges you bought was
• A + (5/6)×A = (6/6)×A + (5/6)×A = ((6+5)/6)×A = (11/6)×A....meaning you bought 11/6^th of that original "certain number"
• Lastly you sold them all at 16 for $6.00, meaning that each orange was sold at $6/16
• We can write this total profit as (6/16)×(11/6)×A = (11/16)×A dollars

So we found out three important terms of how much money was given using A, a "certain number of oranges." The first two terms add up to how much money you paid; but remember, they're not equal to the third term of profit because you did not break even. Your profit is larger than the sum of the other two:

(1/3)×A + (5/24)×A < (11/16)×A

Luckily, we know how much greater the profit is: it's by $3.50, or $7/2. So we will add it to the two terms on the left because we know that that should raise the left side to be equal to the right side:

(1/3)×A + (5/24)×A + 7/2 = (11/16)×A

Now we are ready to solve for A, a "certain number of oranges:"

(8/24)×A + (5/24)×A + 7/2 = (11/16)×A  , multiply the top & bottom to get the denominators the same

((8+5)/24)×A + 7/2 = (11/16)×A

(13/24)×A + 7/2 = (11/16)×A

(13/24)×A - (11/16)×A = -7/2  , 24 = 8×3 & 16 = 8×2

(26/48)×A - (33/48)×A = -7/2  , 48 = 8×2×3, so it's the LCD

((26-33)/48)×A = -7/2

(-7/48)×A = -7/2

A = 24

This is the mysterious "certain number of oranges" at the beginning of the problem, but it's not the number they're asking for. You want to know how many total oranges you bought, which we found out earlier to be:

(11/6)×A = (11/6)×24 = 44

I hope this helps. Have a good weekend!