Algebraic Language

Here we will become more fluent in translating algebraic language to our spoken language and vice versa.

Some important guidance:

1. Be sure to write a key, detailing what symbols mean what using “let”.
2. Write the symbols you have defined mathematically, using what you are told from the question.
3. Then solve the equation, translating the symbols back into words.
4. Make sure your answer is realistic, for example, it is not possible to have -⁵/₃ as an answer if the question was how many people are on a train.

Key Words	Translation
Sum, increased by, more than, plus, added to, total	+
Difference, decreased by, subtracted from, less, minus	−
Product, multiplied by, of, times, twice	*
Quotient, divided by, ratio, per	/ or ÷
Is, total, result	=

Credited with thanks to tohttps://saylordotorg.github.io/text_elementary-algebra/s05-05-applications-of-linear-equatio.html

Key Phrases	Translation
The sum of a number and 7. Seven more than a number.	+ 7
The difference of a number and 7 Seven less than a number. Seven subtracted from a number.	- 7
The product of 2 and a number Twice a number.	2 x
One-half of a number	x  1/2
The quotient of a number and 7.	÷ 7

Solving the equation should be the easy part, as that’s what we have been covering up until now. The hard part is reading what the question says and correctly translating it into algebra.

Question 1

The solution to the above question is repeated with a more detailed explanation below.

Pick any number and apply it to the sentence, this will give us an idea of what the question will look like.

Let us pick 1 as an easy number.

The sum of a number and six more than the number is twelve.

That number is 1, so, the sum of 1 and 6 more than 1 is 12.

We have the sum (addition) of 1, and we have 6 more than one, which we will write like

1 + (6 + 1) is 12

1 + 6 + 1 = 12

8 = 12,

So, the number is not 1, but the point in this question was not to be right, but to get a feel of what the algebra will look like.

Start with a “let” statement, then write the sum of a number and six more than the number is twelve algebraically.

All we have to do for our let statement is say,

Question 2

The solution to the above question is repeated with a more detailed explanation below.

First, start with our “let” statement,

Let n = the unknown number.

Writing in these simple symbols we are familiar with helps us to start seeing the sentence as an algebraic equation.

Anytime you see “and”, it can help to write brackets around the two values the “and” is combining, so,

The difference between (3 times n) and (n that is 4 more than it) is 6

Think about “The difference between” as subtraction, [Say for example you wanted to know “the difference between” a car that was worth $10,000 and a car that was worth $5,000 - you would take the $10,000 and subtract the $5,000]

Question 3

The solution to the above question is repeated with a more detailed explanation below.

When figuring out how to write these questions mathematically, pay attention to the way things are defined. Laura’s money is defined by Steve’s as she has “16 dollars less than Steve”. This hints to us to write our let statement like; Let Laura’s money = Steve’s money - 16. We are not told anything yet specifically about Steve’s money, so we will simply write it as Steve’s money = S

Start with our let statements,

Let the amount of money Steve has at the beginning be:

S

Let the money Laura has be:

S - 16

Now we have our first part, we move onto the part of the question that gives us enough information to solve the problem.

When Steve spends half his money (or times by ½), the amount of money Laura will have is 4 dollars more than Steve.

So, after Steve has spent money we have:

The amount of money Steve has is:

(½)S

The amount of money Laura has is:

S - 16 (remember, she didn't spend any so still has the same)

We also know, that after Steve spends half his money, Laura has 4 dollars more than that.

So, we can also write, the amount of money Laura has as:

Hint; when solving these types of questions, we want to be looking for two different ways of defining the same thing, so we can equate them and solve. In this example, we have two different ways of saying how much money Laura has, (½) + 4 and S - 16