# 1.2 Use the language of algebra  (Page 6/18)

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## Translate an english phrase to an algebraic expression

In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. [link] summarizes them.

Operation Phrase Expression
the sum of $a$ and b
$a$ increased by b
b more than $a$
the total of $a$ and b
b added to $a$
$a+b$
Subtraction $a$ minus b
the difference of $a$ and b
$a$ decreased by b
b less than $a$
b subtracted from $a$
$a-b$
Multiplication $a$ times b
the product of $a$ and b
twice $a$
$a·b,ab,a\left(b\right),\left(a\right)\left(b\right)$

2 a
Division $a$ divided by b
the quotient of $a$ and b
the ratio of $a$ and b
b divided into $a$
$a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}$

Look closely at these phrases using the four operations:

Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.

Translate each English phrase into an algebraic expression: the difference of $17x$ and $5$ the quotient of $10{x}^{2}$ and $7.$

## Solution

1. The key word is difference , which tells us the operation is subtraction. Look for the words of and and t o find the numbers to subtract.

2. The key word is “quotient,” which tells us the operation is division.

This can also be written $10{x}^{2}\text{/}7\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\frac{10{x}^{2}}{7}.$

Translate the English phrase into an algebraic expression: the difference of $14{x}^{2}$ and 13 the quotient of 12 x and 2.

$14{x}^{2}-13$ $12x÷2$

Translate the English phrase into an algebraic expression: the sum of $17{y}^{2}$ and $19$ the product of $7$ and $y.$

$17{y}^{2}+19$ 7 y

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.

Translate the English phrase into an algebraic expression: Seventeen more than y Nine less than $9{x}^{2}.$

## Solution

1. The key words are more than. They tell us the operation is addition. More than means “added to.”
$\begin{array}{}\\ \hfill \text{Seventeen more than}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{Seventeen added to}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill y+17\hfill \end{array}$
2. The key words are less than . They tell us to subtract. Less than means “subtracted from.”
$\begin{array}{}\\ \\ \\ \hfill \phantom{\rule{5em}{0ex}}\text{Nine less than}\phantom{\rule{0.2em}{0ex}}9{x}^{2}\hfill \\ \hfill \phantom{\rule{5em}{0ex}}\text{Nine subtracted from}\phantom{\rule{0.2em}{0ex}}9{x}^{2}\hfill \\ \hfill \phantom{\rule{5em}{0ex}}9{x}^{2}-9\hfill \end{array}$

Translate the English phrase into an algebraic expression: Eleven more than $x$ Fourteen less than $11a.$

$x+11$ $11a-14$

Translate the English phrase into an algebraic expression: 13 more than z 18 less than 8 x .

$z+13$ $8x-18$

Translate the English phrase into an algebraic expression: five times the sum of m and n the sum of five times m and n .

## Solution

There are two operation words— times tells us to multiply and sum tells us to add.

1. Because we are multiplying 5 times the sum we need parentheses around the sum of m and n , $\left(m+n\right).$ This forces us to determine the sum first. (Remember the order of operations.)
$\begin{array}{}\\ \\ \hfill \text{five times the sum of}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\left(m+n\right)\hfill \end{array}$
2. To take a sum, we look for the words “of” and “and” to see what is being added. Here we are taking the sum of five times m and n .
$\begin{array}{c}\hfill \text{the sum}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}\text{five times}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\hfill \\ \hfill 5m+n\hfill \end{array}$

Translate the English phrase into an algebraic expression: four times the sum of p and q the sum of four times p and q .

$4\left(p+q\right)$ $4p+q$

Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than $110,000 Michael Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost$1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her$2.52 per pound, how many pounds of each berry should she use?
$1.38 worth of strawberries +$1.14 worth of blueberries which= $2.52 Leitha how Zaione is it right😊 Leitha lol maybe Robinson 8 pound of blueberries and 22 pounds of strawberries Melissa 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa 8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound Melissa hmmmm...... ? Robinson 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa The question asks how many pounds of each in order for her to have an average cost of$2.52. She needs 30 lb in all so 30 pounds times $2.52 equals$75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has \$4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce