Download translating english to math

TRANSLATING
ENGLISH TO MATH
SJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
• One of the major skills required in
mathematics is the ability to translate a
verbal statement into a mathematical
(variable) expression or equation.
• This ability requires recognizing the verbal
phrases that translate into mathematical
operations.
Addition
•
•
•
•
•
•
Added to
(the sum of)
(the total of)
Increased by
Plus
More than
Note: The sum is the answer to an addition
problem. “The sum of x and y”  (x + y)
Subtraction
•
•
•
•
•
•
Subtracted from
(the difference between)
Less
Decreased by
Minus
Less than
Note: The difference is the answer to a subtraction.
“The difference between x and y”  (x – y)
Multiplication
•
•
•
•
•
Times
The product of
Multiplied by
Of
Twice
Note: The product is the answer to a multiplication
problem. “The product of x and y”  (x)(y)
Division
• Divided by
• The quotient of
• The ratio of
Note: The quotient is the answer to a division
problem. “The quotient of x and y”  x .
y
Power
• The square of
• The cube of
exponent 2
exponent 3
Equals
• Equals
• Is/Are/Was/Were
• Amounts to
Note: These lists are not complete.
Translate the following verbal
expressions into variable expressions:
Ex: y added to sixteen
What is the operation?
Addition
What is being added?
y and 16
Write the expression:
y + 16
Ex: the sum of b and eight
What is the operation?
Addition
What is being added?
b and 8
Write the expression:
(b + 8)
Ex: the total of four and m
What is the operation?
Addition
What is being added?
4 and m
Write the expression:
(4 + m)
Ex: w increased by fifty-five
What is the operation?
Addition
What is being added?
w and 55
Write the expression:
w + 55
Ex: g plus twenty
What is the operation?
Addition
What is being added?
g and 20
Write the expression:
g + 20
Ex: nineteen more than K
What is the operation?
Addition
What is being added?
19 and K
Write the expression:
K + 19
Can also by written as:
19 + K
Ex: n subtracted from two
What is the operation?
Subtraction
What is being subtracted?
n and 2
Write the expression (careful!):
2–n
Subtraction does not possess
the commutative property so
order is important. There is a
difference between what goes
in front of the subtraction sign
(the minuend) and what goes
after (the subtrahend).
Ex: the difference of q and three
What is the operation?
Subtraction
What is being subtracted?
q and 3
Write the expression (careful!):
(q – 3)
Ex: r less twelve
What is the operation?
Subtraction
What is being subtracted?
r and 12
Write the expression (careful!):
r – 12
Ex: seventeen decreased by m
What is the operation?
Subtraction
What is being subtracted?
17 and m
Write the expression (careful!):
17 – m
Ex: w minus 3
What is the operation?
Subtraction
What is being subtracted?
w and 3
Write the expression (careful!):
w–3
Ex: nineteen less than d
What is the operation?
Subtraction
What is being subtracted?
19 and d
Write the expression (careful!):
d – 19
Subtraction does not possess
the commutative property so
order is important. There is a
difference between what goes
in front of the subtraction sign
(the minuend) and what goes
after (the subtrahend).
Ex: nine times c
What is the operation?
Multiplication
What is being multiplied?
9 and c
Write the expression:
9c
Ex: the product of negative six and b
What is the operation?
Multiplication
What is being multiplied?
-6 and b
Write the expression:
-6b
Ex: five multiplied by a number
What is the operation?
Multiplication
What is being multiplied?
5 and n
Write the expression:
5n
Ex: fifteen precent of the selling price
What is the operation?
What is being multiplied?
Write the expression:
Multiplication
15% and p
.15p
Ex: twice a number
What is the operation?
What is being multiplied?
Write the expression:
Multiplication
2 and n
2n
Ex: the square of a number
What is the operation?
Multiplication
What is being multiplied?
n and n
Write the expression:
n2
Ex: the cube of a number
What is the operation?
Multiplication
What is being multiplied?
n and n and n
Write the expression:
n3
Ex: four divided by y
What is the operation?
Division
What is being divided?
4 and y
Write the expression (careful!):
4
y
Ex: the quotient of the opposite of n and nine
What is the operation?
Division
What is being divided?
-n and 9
Write the expression (careful!):
-n
9
Ex: the ratio of eleven and p
What is the operation?
Division
What is being divided?
11 and p
Write the expression (careful!):
11
p
Division does not possess the
commutative property so order is
important. There is a difference
between what goes in front of the
division sign/on top (the dividend)
and what goes after the division sign
/on the bottom (the divisor).
Ex: nine increased by the quotient of t and
five
What operation(s)?
Addition and Division
Take it word for word to translate:
nine increased by the quotient
9
+
t
 
5 
of t and five
Ex: the product of a and the sum of a and
thirteen
What operation(s)?
Multiplication and Addition
Take it word for word to translate:
The product of a and the sum of a and thirteen
( a )( a + 13 )
Ex: the quotient of nine less than x and
twice x
What operation(s)?
Division & Subtraction &
Multiplication
Take it word for word to translate:
the quotient of 9 less than x and twice x



x - 9
2 x



Translate into a variable expression and
then simplify.
Identify any variables used.
Ex: a number added to the product of five and
the number
“a number”, “the number”  let n = a number
What operation(s)? Addition and Multiplication
a number added to the product of 5 and the number
n
+
( 5 )( n )
= 6n when simplified (combine like terms)
Ex: a number minus the sum of the number
and fourteen
“a number”, “the number”  let n = a number
What operation(s)?
Subtraction and Addition
a number minus the sum of the number and 14
_
n
( n + 14 )
= n – n – 14
(removing parentheses)
= – 14
(combining like terms)
Ex: Twice the quotient of four times a number
and eight
“a number” 
let n = a number
What operation(s)?
Multiplication and Division
twice the quotient of 4 times a number and 8
2
4n 


8 
8n

8
= n
(Simplifying)
(Simplifying)
Equals
•
•
•
•
•
Equals
Is/Are/Was/Were
Amounts to
The results is
To obtain
Note: These lists are not complete.
Write a variable expression.
Identify any variables used.
Ex: The sum of two numbers is 18.
Express the numbers in terms of the same
variable.
* If the sum of two numbers is 9 and the first
number is 5, what is the second?
4 How did you get that? Subtract: 9 – 5 = 4
* If the sum of two numbers is 17 and the first
number is 12, what is the second?
5 How did you get that? Subtract: 17 – 12 = 5
Back to the example:
Ex: The sum of two numbers is 18.
Let n = first number.
Then the second number is found by
subtracting:
18 – n = second number
Check: Do n and 18 – n sum to 18?
n + (18 – n)
= n + 18 – n
= 18
Translate the English sentences into equations
and solve. Identify any variables used.
Ex: The sum of five and a number is three.
Find the number.
What are we looking for? The number = n
Translate word-for-word:
The sum of 5 and a number is 3
(__
5 + __)
n
= 3
Now solve 5 + n = 3 by subtracting 5 from both sides
-5
-5
n=-2
Ex: The difference between five and twice a
number is one. Find the number.
What are we looking for? The number = n
Translate word-for-word:
The difference between 5 and twice a number is 1
(______
5
– ______)
2n
= 1
Now solve 5 – 2n = 1
– 2n = - 4
n=2
Subtract 5 from both sides
Divide both sides by - 2
Ex: Four times a number is three times the difference
between thirty-five and the number. Find the number.
What are we looking for? The number = n
Translate word-for-word:
Four times a number is three times
the difference between 35 and the number
n
35
(
_____
_____)
4n = 3
Solve 4n = 3(35 – n)
Simplify – distribute 3
4n = 105 – 3n
Collect like terms (add 3n to both sides)
7n = 105
Divide both sides by 7
n = 15
Ex: The sum of two numbers is two. The
difference between eight and twice the smaller
number is two less than four times the larger.
Find the two numbers.
What are we looking for? Two numbers:
Let s = smaller number. Then 2 – s = larger number.
The difference between 8 and twice the smaller
is two less than four times the larger
8
2 s
4 (2 – s) - 2
( ______
______
) = _________
Solve 8 – 2s = 4(2 – s) – 2
8 – 2s = 8 – 4s – 2
8 – 2s = 6 – 4s
Simplify
Combine like terms
Collect like terms
(add 4s to both sides)
8 + 2s = 6
Collect like terms
(subtract 8 from both sides)
2s = - 2
Divide both sides by 2
s = - 1  smaller number is -1
Therefore, 2 – s = 2 – ( - 1) = 3 the larger number is 3
Ex: A college employs a total of
600 teaching assistants (TA) and
research assistants (RA). There
are three times as many TAs as
RAs. Find the number of RAs
employed by the university.
What are we looking for? The number of RAs = r
The number of TAs =
600 - r
There are three times as many TAs as RAs
So are there more TAs or RAs? TAs
How many TAs? 600 – r
The number of TAs is 3 times the number of RAs
600 - r
Solve 600 – r = 3r
600 = 4r
= 3r
(add r to both sides)
(divide both sides by 4)
150 = r  150 RAs and 3r = 3(150) = 450 TAs
Ex: A wire 12 ft long is cut into
two pieces. Each piece is bent
into the shape of a square. The
perimeter of the larger square is
twice the perimeter of the
smaller square. Find the
perimeter of the larger square.
What are we looking for?
The perimeter of the larger square
What do we know?
We will form 2 squares using 2 pieces of wire
The pieces are from cutting a single piece in two:
How long is each piece of wire?
Let s = length of the shorter piece 
Then, 12 – s = length of the longer piece
since we start with a 12 ft wire.
Now, take the smaller wire and bend it into a square
What is the perimeter of the smaller square?
s since the shorter piece is of length s
What is the perimeter of the larger square?
12 – s since the longer piece is of length 12 - s
Now what?
Use the information, translate into an equation:
perimeter of the larger square is
twice the perimeter of the smaller square
12 – s = 2 s
Solve 12 – s = 2s
Add s to both sides
12 = 3s
Divide both sides by 3
4=s
Therefore the shorter piece is 4 ft  the smaller
square has perimeter 4 ft.
Have we answered the question asked?
No. We want to find the perimeter of the larger
square. So, 12 – s = 12 – 4 = 8
8 ft is the perimeter
of the larger square.