Algebra word problems | Arithmetic & algebra | Quantitative reasoning

Translating words to algebra

Algebra word problems generally have simpler math than regular algebra problems. The only catch is that the equation is presented to you in a sentence instead of numeric values. However, with a few quick tips, these problem types will seem like a gift since they tend to be simple once you understand the core concept.

Here are some equivalencies that you can use to translate the next few sentences into equations. Sometimes the word “is” is not explicitly stated. In this case, just have to imagine where to split the equation in a way that makes one side equal to the other.

Word	Algebraic meaning
is	$=$
more than	$+$
less than	$-$
twice $a$	$2 a$
half of $a$	$a /2$
$x$ percent of $a$	$(x /100) a$

Examples:

Tim’s age is twice Sharon’s age.
Garden B is half the size of Garden A.
There are 12 more students in Classroom X than in Classroom Y
Nigeria is 15% of the population of Africa

Can you figure out their corresponding equations?

(spoiler)

$T = 2 S$
$B = A /2$
$X = 12 + Y$
$N = (15/100) A$

The combined work equation

Some word problems require you to solve for the total time it takes for multiple people/machines to complete a job when working together. The combined work equation is a shortcut to solving these questions. In the equation below, A and B represent the time it takes entity A and entity B to complete their jobs when they work alone. The solution is the time it takes to complete a job when working together.

$(A B) / (A + B)$

For example:

A SuperPrinter machine can print 1,000 T-shirts in an hour. How long would it take two SuperPrinter machines to print 1,000 T-shirts total working together?

We can replace both A and B with the number 1 because it takes both machines one hour to complete the one job, i.e. print 1000 t-shirts.

$(A B) / (A + B) (1 \times 1) / (1 + 1) 1/2$

Thus, it would take them both half an hour to print 1,000 t-shirts.

Sidenote

Time formats

If the final answer has a decimal and you are required to answer in hours and minutes, simply multiply the decimal part by 60 to find the minutes.

For example, you could express 2.4 hours as $2$ hours and $.4 \times 60 = 24$ minutes,

Sentences with many variables in the answer choices

Some word problems give you a scenario with lots of variables and no real numbers in both the question and answer choices. It’s your job to pick numbers for the variables and determine which answer choice makes sense based on the numbers you have chosen. For example:

Country A’s population is x percent of Country B’s population. Which of the following answers would represent x in terms of A and B?

A. $A / B$
B. $(A / B)^{2}$
C. $A + B$
D. $100 (A / B)$
E. $100 (B / A)$

(spoiler)

Answer: D.

Plug in values for Country A and B that intuitively make sense. Let’s say Country A has 5 people and Country B has 10 people. That would mean Country A is 50% of Country B. Just plug in 5 for A and 10 for B into each answer choice until you have found a value that equals 50. Because the question asks for the value of x, you just need the number that would go before the percent sign (which is 50). Only answer Choice D (100(A/B)) works. Answer A is incorrect because it amounts to .5, not 50. Remember Country A is 50% of Country B, not .5%.