Here we will become more fluent in translating algebraic language to our spoken language and vice versa.
Some important guidance:
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 | / |
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. | x + 7 |
Seven more than a number. |
The difference of a number and 7. | x - 7 |
Seven less than a number. | |
Seven subtracted from a number. | |
The product of 2 and a number. | 2 x |
Twice a number. | |
One-half of a number. | 1 2 x |
The quotient of a number and 7. | x / 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.
The sum of a number and six more than the number is twelve. What is the number?
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.
All we have to do for our let statement is say,
Let n be the unknown number,
Now we create our equation,
The sum of a number and six more than the number is twelve
Becomes the sum of n and six more than n is twelve,
In these questions, pay attention to anytime you see “and”, it can help to write brackets around the two values the “and” is combining, so,
The sum of (n) and (six more than n),
Note, this does not mean n and 6, but (n) and (n + 6),
So, the equation will look like, the sum of n and n + 6 which is n + (n + 6) is 12, or,
n + (n + 6) = 12
n + n + 6 = 12
2n + 6 = 12
2n + 6 - 6 = 12 - 6
2n = 6
(2n)/ 2 = (6) / 2
The difference between three times a number and a number that is four more than it is six. Write an equation to represent this using a “let” statement and then solve that equation for the unknown number.
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]
(3 times n) - (n that is 4 more than it) is 6,
Put in our operative symbols we can see,
Note how having brackets around the ... - (n + 4) ... makes a difference compared to ... - n + 4 ...
3n - (n) - (4) = 6,
2n - 4 = 6
2n -4 + 4 = 6 + 4
2n = 10
(2n)/2 = (10)/2
n = 5
At the current moment, Trevor’s father (Ray) is 20 years older than him, in 5 years Ray will be twice Trevor’s age. How old is Trevor now?
Begin by having our let statements for now, and our let statements in 5 years,
Now:
Let Trevor’s age = T
Let Ray’s age = Trevor’s age plus 20 = T + 20
In Five years:
Trevor’s age = T + 5
Let Ray’s age = (T + 5) + 20 = T + 25
We also, know, that in 5 years,
Ray’s age will be Twice Trevor’s age (after 5 years), or,
Ray’s age = 2(T +5),
We now have to different expressions that equal Ray’s age
Ray’s age = 2(T +5)
Ray’s age = T + 25
We can equate these,
2(T + 5) = T + 25
Expand and solve,
2T + 10 = T + 25
(2T + 10) - T = (T + 25) - T
2T - T + 10 = T - T + 25
T + 10 = 0 + 25
T + 10 - 10 = 25 - 10
T = 15,
Therefore, Trevor’s age now is 15.
Laura has 16 dollars less than Steve. Steve then spends half of his money and Laura spends none, Laura will have 4 dollars more than Steve. How much money did they have at the beginning?
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, she has “16 dollars less than Steve”. This hints 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]
So,
(½)S + 4 = S - 16,
Additive Property,
((½)S + 4) + 16= (S - 16) + 16
(½)S + 20 = S
Additive Property again,
(½)S + 20 - (½)S = S - (½)S
[S - (½)S = (½)S (i.e. 1 - ½ = ½)]
20 = (½)S
Multiplicative Property, divide both sides by (½) or the same as multiply both sides by 2,
(20)2 = ((½)S)2
40 = S,
Now we look back at our original statements, as just saying S = 40 does not answer the question.
[Let Steve’s money be S,
Laura's money be S - 16,]
So, the original amounts of money,
Steve had 40 dollars,
Laura had 40 - 16 dollars, so, 24 dollars
This next question will involve percentages and money. Be aware that when you are dealing with a percentage you will need to convert it to a fraction in order to work with it mathematically, and if the answer is asking for a percentage, be sure to convert that fraction back to a percentage. For money, make sure that you round to the nearest decimal point.
If a coat costs $84.40 including a tax of 5½%, what is the cost of the hat without tax?
First, convert the percentage to a fraction,
5½% is 5.5%,
100% is equal to 1.00 as a fraction
So, 5.5% is equal to 0.055,
Our final cost of $84.40 is calculated by taking the original price, say c, plus the tax times the original cost of the coat,
So, $84.40 = c + c(0.055)
c + c (0.055) can be written as,
1c + c (0.055) which equals
(1.055) c
So,
$84.40 = (1.055) c, now divide both sides by 1.055,
$84.40 / 1.055 = (1.055) c / 1.055
$80 = c,
So, the cost without tax was $80.