Solved Examples: Translating Word Problems to Algebraic Expressions

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.

For SAT Students
Any question labeled SAT-C is a question that allows a calculator.
Any question labeled SAT-NC is a question that does not allow a calculator.

For JAMB Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE:FM is a question for the WASSCE Further Mathematics/Elective Mathematics

For GCSE and Malta Students
All work is shown to satisfy (and actually exceed) the minimum for awarding method marks.
Calculators are allowed for some questions. Calculators are not allowed for some questions.

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Translate each word problem from English to Math.
Use appropriate variables as applicable.
Do not solve.

(1.) Translate from English to Math.
For:
the first term of an arithmetic sequence, use the variable: a
the common difference, use the variable: d
all other variables, use the variable: n

(a.) 5 less than twice than a number
(b.) The difference between the square of a number and four times the number
(c.) 5 times the square of a number
(d.) The 8th term of an arithmetic sequence whose first term is 15

English	Math
5 less than twice than a number	Twice the number = 2n 5 less than 2n is: 2n − 5
The difference between the square of a number and four times the number	Square of the number = n² Four times the number = 4n Difference between n² and 4n is: n² − 4n
5 times the square of a number	Square of the number = n² 5 times n² is: 5n²
The 8th term of an arithmetic sequence whose first term is 15	The nth term of an arithmetic sequence = a + d(n − 1) n = 8 a = 15 The 8th term of an arithmetic sequence whose first term is 15 is: 15 + d(8 − 1) 15 + 7d

(2.) Translate from English to Math.
(a.) If the number of professors in a college is P and the number of students S, and there are 12 times as many students as professors, write an algebraic equation that shows the relationship.

(b.) If g is the number of girls in a class and b the number of boys and if there are thirteen more girls g than boys b in a class, write an algebraic equation that shows this relationship.

(a.) 12 times as many students as professors ⇒ S = 12P

(b.) Thirteen more girls g than boys b in a class ⇒ g = 13 + b

(3.) ACT Which of the following mathematical expressions is equivalent to the verbal expression "A number, $x$, squared is $39$ more than the product of $10$ and $x$"?

$ F\:\: 2x = 39 + 10x \\[3ex] G.\:\: 2x = 39x + 10x \\[3ex] H.\:\: x^2 = 39 - 10x \\[3ex] J.\:\: x^2 = 39 + x^{10} \\[3ex] K.\:\: x^2 = 39 + 10x \\[3ex] $

A number, $x$ means $x$

squared means $x^2$

is means $=$

product of $10$ and $x$ means $10 * x = 10x$

$39$ more than the product of $10$ and $x$ means $39 + 10x$

A number, $x$, squared is $39$ more than the product of $10$ and $x$ means $x^2 = 39 + 10x$

(4.) CSEC Write the following statement as an algebraic expression.
The sum of a number and its multiplicative inverse is five times the number.

Let the number be $p$
Multiplicative inverse of $p = \dfrac{1}{p}$

$ p + \dfrac{1}{p} = 5p $

(5.) Nahum bought y children's admission tickets for $2 each and x adult's admission tickets for $7 each.
Write an algebraic expression for the total amount spent by Nahum.

$y$ children's admission tickets for $\$2$ each = $y * 2 = 2y$

$x$ adult's admission tickets for $\$7$ each = $x * 7 = 7x$

$ Total = 2y + 7x $

(6.) Samson's debt is seven less than half of David's debt.
If d represents David's debt, write an expression for Samson's debt.

David's debt = d
Half of David's debt = $\dfrac{1}{2} * d = \dfrac{d}{2}$

Seven less than half of David's debt = $\dfrac{d}{2} - 7$

Therefore, Samson's debt = $\dfrac{d}{2} - 7$

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