# 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 .

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

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be? 74.25 × .16 then get the total and that will be your tip David$74.25 x 0.16 = $11.88 total bill:$74.25 + $11.88 =$86.13
ericka
yes and tip 16% will be $11.88 David what is the shorter way to do it Cesar Reply Priam has dimes and pennies in a cup holder in his car. The total value of the coins is$4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne