Algebra word problems | Arithmetic & algebra | Quantitative reasoning

Translating words to algebra

Algebra word problems usually involve simpler calculations than many “pure” algebra problems. The main difference is that the equation is written in words instead of numbers and symbols. Your job is to translate the sentence into an equation.

A helpful trick is to look for the point where the sentence is telling you that two expressions are equal. Sometimes the word “is” isn’t stated directly. When that happens, decide where the sentence naturally splits into a left side and a right side that should be equal.

Here are some common translations you can use.

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 ask how long it takes for two people or machines to complete one job when they work together.

The combined work equation is a shortcut for these problems. In the equation below:

$A$ is the time it takes entity A to complete the job working alone.
$B$ is the time it takes entity B to complete the job working alone.
The result is the time it takes them to complete the job 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?

Each machine takes 1 hour to complete the job alone, so $A = 1$ and $B = 1$ .

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

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

Sidenote

Time formats

If your final answer is a decimal number of hours and you need to write it in hours and minutes, multiply the decimal part by 60 to convert it to 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 describe a situation using variables (not actual numbers) in both the question and the answer choices. In these problems, you can choose simple numbers for the variables and test the answer choices to see which one matches the situation.

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.

Pick values for Country A and B that are easy to work with. For example, let $A = 5$ and $B = 10$ . Then Country A is 50% of Country B, so $x = 50$ .

Now plug $A = 5$ and $B = 10$ into each answer choice and see which one equals 50.

Only choice D works:

$100 (A / B) = 100 (5/10) = 50$

Choice A is incorrect because $A / B = 5/10 = 0.5$ , which represents the decimal form of the ratio, not the percent. The question asks for the number that goes before the percent sign.

Translating words to algebra

Here are some common translations you can use.

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 ask how long it takes for two people or machines to complete one job when they work together.

The combined work equation is a shortcut for these problems. In the equation below:

$A$ is the time it takes entity A to complete the job working alone.
$B$ is the time it takes entity B to complete the job working alone.
The result is the time it takes them to complete the job 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?

Each machine takes 1 hour to complete the job alone, so $A = 1$ and $B = 1$ .

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

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

Sidenote

Time formats

If your final answer is a decimal number of hours and you need to write it in hours and minutes, multiply the decimal part by 60 to convert it to 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

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.

Pick values for Country A and B that are easy to work with. For example, let $A = 5$ and $B = 10$ . Then Country A is 50% of Country B, so $x = 50$ .

Now plug $A = 5$ and $B = 10$ into each answer choice and see which one equals 50.

Only choice D works:

$100 (A / B) = 100 (5/10) = 50$

Choice A is incorrect because $A / B = 5/10 = 0.5$ , which represents the decimal form of the ratio, not the percent. The question asks for the number that goes before the percent sign.