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Word Problems with Fractions

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
Summary
Word problems with fractions become much easier when key translation rules are applied consistently: 'is' means equals, and 'of' means multiply. The fastest path to correct ACT-style solutions is to turn verbal relationships into equations, assign variables clearly, and isolate the quantity the question actually asks for.
  • Start by translating fraction language into math using two core rules: 'is' means '=', and 'of' means multiplication.
  • Use these rules in straightforward problems such as finding a fraction of a number by multiplying the fraction and the whole, then simplifying before multiplying.
  • For real-world situations, assign variables to each quantity, translate the statement into an equation, substitute any known values, and solve by multiplying by the reciprocal when needed.
  • In multi-step comparison problems, write separate equations for the same quantity so different quantities can be related through it.
  • Focus on the exact relationship the question asks for, eliminate irrelevant variables when possible, and express the final answer as the requested fraction.
Chapters
00:00
Core Translation Rules for Fraction Word Problems
01:01
Using Variables and Reciprocals in Real-World Problems
02:02
Comparing Multiple Quantities Through One Shared Value

Q: In the last problem, I would have liked to see how the N was isolated. I did not follow the math on the reciprocal to isolate N.

A: Not to worry! Let's take a look: 

Cathy's Salary is 3/7 of Nora's salary and (Cathy's salary) is also 5/4 of Teresa's salary.

C = (3/7)N

C is also equal to (5/4)T

So we can set:

(3/7)N = (5/4)T

Now, I isolate N by multiplying both sides of the equation by (7/3), the reciprocal of 3/7.

(7/3)(3/7)N = (7/3)(5/4)T

(7/3)(3/7) = 1 so (7/3)(3/7)N just equals N.

N = (7/3)(5/4)T =

(35/12)T

Q: The lesson doesn't really explain why we multiply by the reciprocal to get the answer in the second problem. I am very confused.

A: Happy to help! This is just an extension of one of the rules of working with algebraic expressions: when trying to isolate a variable, you can perform a number of operations to both sides of an equation, but they have to be done to both sides.

So, for example, say I have this equation:

x/2 = 4

If I want to get x alone, I multiply it by 2. But I also have to do the same to the other side:

x/2 = 4 2x/2 = 2 * 4 x = 8

Now, let's try that with this equation:

2x/7 = y

First, I multiply out that 7:

(7 * 2x)/7 = 7 * y 2x = 7y

Then, I divide out the 2:

2x/2 = 7y/2 x = 7y/2

Look again at my original and final equations:

2x/7 = y x = 7y/2

In order to isolate x, I multiplied y by the reciprocal of the fraction on the left side. That's just the easier, faster way to do it. It's shorthand for the same process.

In the lesson video, there were two cases of expressions like that:

300 = 2r/7 7 * 300 = 2r 7 * 300 / 2= r

And here's the other:

c = 3n/7  7c = 3n 7c/3 = n