Download Equation Building - Answer Explanations

Equation Building
The most effective way of approaching many word problems is to find the equation or
equations that are implied by the words. To do so effectively, you must know what operations
are implied by certain common words.
a number x (the unknown to be solved for)
of multiplication
out of division
per usually division but sometimes multiplication, depending on the situation/context
is equals
the same equals
how many x (the unknown to be solved for)
sum addition
difference subtraction
product multiplation
times multiplication
quotient division
exceeds addition
greater than addition or
less than subtraction or
1) Three times a number is 75.
3x = 75
2) 75% of a number is 60.
.75 x = 60
3) You had a dozen donuts. You ate 5/6 of them. How many donuts did you eat?
12 • 5/6 = x
4) There are 12 donuts in a box. If you have three boxes of donuts, how many donuts do you
have?
12 • 3 = x
5) You drove 210 miles in three hours. How many miles per hour did you average?
210/3 = x
6) The length of a rectangle exceeds twice the width by 10. If the perimeter of the rectangle is
44 feet, what is the length?
l = 2w + 10
2l + 2w = 44
Once you have the initial equations, use substitution to get 2(2w + 10) + 2w = 44.
7) The height of a triangle is 5 less than three times the length of its base. If the triangle’s area
is 14 square inches, what is the height of the triangle?
h = 3b – 5
½b • h = 14
Once you have the initial equations, use substitution to get ½b(3b – 5) = 14. After solving for b,
plug into either of the initial equations to find h.
8) Three times a number is 2 less than 4 times the number.
3x = 4x – 2
9) What number is 20% greater than 55?
x = 55 + .2 • 55 or x = 1.2 • 55
10) 98 is 40% greater than what number?
98 = x + .4x or 98 = 1.4x
In building linear equations, it is important to keep in mind that the y-intercept is always
equal to the fixed amount and the slope is equal to the variable or “per something” amount.
11) At the beginning of year one of a study, the height of a tree is 15 feet. If the tree grows at a
rate of 3 feet per year, what is the height of the tree after x years?
h = 3x + 15
12) One airplane begins descending from an altitude of 21,000 feet at a rate of 1,000 feet per
minute. At the same time the first airplane begins its descent, another airplane takes off and
gains altitude at a rate of 2,000 feet per minute. After how many minutes will the two planes
be at the same altitude?
a = 21,000 – 1,000m
a = 2,000m
Because the questions ask when the two planes will be at the same altitude, we know that the
“a” in each equation must be equal, so you can get the following equation via substitution:
21,000 – 1,000m = 2,000m