Algebraic Expressions

Scott Solomon

Everyday Language  —>  Math Action

Basics
An algebraic expression is often a word problem.
A word problem requires you to translate Everyday Language into Math Action.

Addition language can flip-flop:  y + 5 is the same as 5 + y
Subtraction language cannot flip-flop:  y – 5 is not the same as 5 – y
Multiplication language can flip-flop:  5y is the same as y5
Division language cannot flip-flop:  \bf\displaystyle\frac{5}{y} is not the same as \bf\displaystyle\frac{y}{5}

For the sake of order of operations, it is better to set up division language in \bf\displaystyle\frac{x}{y} form rather than x÷y form.

When a number is immediately adjacent to a letter or parenthesis, an invisible multiplication sign is in force.
3y = 3 × y
3(y) = 3 × y
3(5) = 3 × 5


Dictionary – Algebraic Expressions
The following table lists Everyday words under the Math actions that they represent.

Addition Subtraction Multiplication Division
add subtract multiply divide
sum difference product quotient
plus minus times share
more less compound split
increase decrease augment fit
greater fewer
Expressions

 

Question
What is 4 more than y?

Answer
y + 4
or
4 + y

Answer Process
more means addition
addition language can flip-flop


Question
The number of dogs d in our neighborhood has increased by 7.  Write an expression that represents this increase.

Answer
d + 7
or
7 + d

Answer Process
increase means addition
addition language can flip-flop


Question
What is 4 less than y?

Answer
y – 4
(but not 4 – y)

Answer Process
less means subtraction
subtraction language cannot flip-flop


Question
The number of cats c in our neighborhood has decreased by 7.  Write an expression that represents this decrease.

Answer
c – 7
(but not 7 – c)

Answer Process
decrease means subtraction
subtraction language cannot flip-flop


Question
What is 4 multiplied by y?

Answer
4y
or
y4

Answer Process
multiply means multiplication
multiplication language can flip-flop


Question
The number of birds in our neighborhood is 7 times the number of birds b in any other neighborhood.  Write an expression that represents the number of birds in our neighborhood.

Answer
7b
or
b7

Answer Process
times means multiplication
multiplication language can flip-flop


Question
What is 4 divided by y?

Answer
 \bf\displaystyle\frac{4}{y}
(but not \bf\displaystyle\frac{y}{4})

Answer Process
divide means division
division language cannot flip-flop


Question
The number of hamsters h in our neighborhood will be shared among a total of 7 neighborhoods (including our own).  Write an expression that represents the number of hamsters that each neighborhood will share.

Answer
 \bf\displaystyle\frac{h}{7}
(but not \bf\displaystyle\frac{7}{h})

Answer Process
share means division
division language cannot flip-flop


Question
The number of cats c in our neighborhood is 7 more than 5 times the number of dogs d.  Write an expression that represents the number of cats in relation to the number of dogs in our neighborhood.

Answer
c = 5d + 7
or
c = 7 + 5d

Answer Process
number of cats relates to 5 times number of dogs → 5d
cats are 7 more than 5 times dogs → 5d + 7
c = 5d + 7     or     c = 7 + 5d


Question
The number of hamsters h in our neighborhood is 4 less than number of birds b in our neighborhood split among a total of 7 neighborhoods (including our own).  Write an expression that represents the number of hamsters in relation to the number of birds in our neighborhood.

Answer
h = \bf\displaystyle\frac{b}{7} – 4
(but not h = 4 – \bf\displaystyle\frac{b}{7})

Answer Process
number of hamsters relates to number of birds split among 7 neighborhoods → \bf\displaystyle\frac{b}{7}
hamsters are 4 less than birds split among 7 neighborhoods → \bf\displaystyle\frac{b}{7} – 4
h = \bf\displaystyle\frac{b}{7} – 4


Question
A full partner in a law firm charges $300.00 an hour.  A junior associate in the law firm charges $50.00 an hour.  The law firm bills a client for a case in which the junior associate worked 10 more hours than the full partner.  Write an expression in terms of hours h worked.

Answer
300h + 50(h+10)

Answer Process
full partner’s hours → h
full partnerr’s part of bill → 300h
junior associate’s hours → h + 10
junior associate’s part of bill → 50(h + 10)
putting it all together —> 300h + 50(h + 10)


Question
A full partner in a law firm charges $300.00 an hour.  A junior associate in the law firm charges $50.00 an hour.  The law firm bills a client for a case in which the junior associate worked 10 more hours than the full partner.  Write an expression that represents the total bill T in terms of hours h worked.

Answer
T = 300h + 50(h+10)

Answer Process
full partner’s hours → h
full partner’s part of bill → 300h
junior associate’s hours → h + 10
junior associate’s part of bill → 50(h + 10)
total bill → T
putting it all together —> T = 300h + 50(h + 10)

 

Question
A full partner in a law firm charges $300.00 an hour.  A junior associate in the law firm charges $50.00 an hour.  The law firm bills a client a total of $6000.00 for a case in which the junior associate worked 10 more hours than the full partner.  Write an expression that represents the total bill in terms of hours h worked.

Answer
300h + 50(h+10) = 6000

Answer Process
full partner’s hours → h
full partner’s part of bill → 300h
junior associate’s hours → h + 10
junior associate’s part of bill → 50(h + 10)
total bill → 6000
putting it all together —> 300h + 50(h + 10) = 6000


Practice – Questions
1.  What is 6 more than x?

2.  What is 6 less than x?

3.  What is 6 times x?

4.  What is x divided by 6?

5.  What is the difference between 6 times x and 7 divided by y?

6.  The number of bottles in a drugstore is 4000 less than 3 times the number of pills.  Write an expression that represents the number of bottles b in relation to the number of pills p in the drugstore.

7.  The number of jars in the local food bank is 7 more than the number of cans in the local food bank shared among a total of 9 food banks (including the local food bank).  Write an expression that represents the number of jars j in relation to the number of cans c in the local food bank.

8.  A surgeon charges $2500.00 per service performed.  A physician’s assistant charges $100.00 per service performed.  A patient receives a bill in which a physician’s assistant performed 5 times as many services as a surgeon.  Write an expression in terms of services s performed.

9.  A hospital employs 100 registered nurses and 250 nurse’s assistants.  A nurse’s assistant earns $2140.00 a month less than a registered nurse.  Let y represent total nursing salaries in any given month and x represent the salary of a registered nurse in any given month.  Write an expression that represents y in terms of x.

10.  At the candy store, a customer purchased x buttercreams at $0.25 each, the same number of toffees as buttercreams at $0.62 each, and three times as many nougats as buttercreams at $0.88 each.  The customer spent a total of $19.64.  Write an expression that represents the customer’s purchase in terms of x.


Practice – Answers
1.  x + 6     or     6 + x

2.  x – 6

3.  6x     or     x6

4.  \bf\displaystyle\frac{x}{6}

5.  6x – \bf\displaystyle\frac{7}{y}

6.  b = 3p – 4000

7.  j = \bf\displaystyle\frac{c}{9} + 7     or     j = 7 + \bf\displaystyle\frac{c}{9}

8.  2500s + 100(5s)

9.  y = 100x + 250(x – 2140)

10.  0.25x + 0.62x + 0.88(3x) = 19.64

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