How to Begin:
1.
2.
3.
4.
5.
Read the question.
Identify type of problem.
Set up approach to solving &
write equations. (Draw a
picture to help.)
Do the math. What are you
trying to solve for?
Read the final question; does
the answer make sense in the
context of the problem? (i.e.
distance is never negative)
Distance
Word Clues
0 Distance:
0 Miles, kilometers
0 Rate:
0 miles per hour,
0 Time:
0 minutes, seconds,
hours
Method
0𝑫=𝒓∙𝒕
D=
Person/
Vehicle A
Person/
Vehicle B
r
t
Geometry
Word Clues
0 Shapes:
0 Rectangular
0 Triangular
0 Circular
0 Trapezoid
0 Cylinder
0 Box
Method
Know the formulas!
(Make flashcards!)
0 Rectangle:
0 𝐴 = 𝑙𝑤
0 𝑃 = 2𝑙 + 2𝑤
0 Triangle:
1
2
0 𝐴 = 𝑏ℎ
0 𝑃 = 𝑠1 + 𝑠2 + 𝑠3
0 Circle:
0 𝐴 = 𝜋𝑟 2
0 𝑃 = 𝜋𝑑 = 2𝜋𝑟
Mixture Problems
Word Clues
Method
0 Percentages of
measurements in
weight, volume,
capacity
0 mixture
0 Solutions, nuts,
coffee, candies
Amount
Item
Type A
Item
Type B
Mixture
%
Total
Simple Interest
Word Clues
Method
0 Interest, rate
I
0 Principle invested
in dollars
0 Amount of money
0 Time in years
=
P
×
r
Account
A
Account
B
Total
Amount
0 𝐴 = 𝑃 + 𝑃𝑟𝑡
0 where 𝐴 = final amount
×
t
Work –Ratio Problems
Word Clues
0 Amount of activity
performed in unit
of time
0 two or more
participants
0 Question of ratio to
complete job
“together”
Method
0 Use ratios!
1 1 1
+ =
𝑎 𝑏 𝑥
Other Types
0 Given formula(s) → Plug it in.
0 The sum or difference of two or more numbers → See
translation table handout.
0 Interpretation → Know your terminology.
Question #1
0 A car leaves Salt Lake City traveling at 70mph. An
hour later, a second car leaves Salt Lake City following
the first car, speeding at 80 mph. How many hours
after leaving Salt Lake City will it take the second car
to catch up to (pass, overtake) the first car?
0 HINT: Both cars leave Salt Lake. When the 2nd car catches up to, passes, or overtakes the 1st
car, we understand that they have traveled the same distance.
Solution #1
D
r
t
Car 1
70 ∙ 𝑡
70mph
𝑡
Car 2
80(𝑡 − 1)
80mph
𝑡−1
So we have
Car 1: 𝐷1 = 70 ∙ 𝑡
Car 2: 𝐷2 = 80(𝑡 − 1)
And since the distance is equal for both cars we can set the equations equal to
each other.
𝐷1 = 𝐷2
70 ∙ 𝑡 = 80(𝑡 − 1)
𝑡=7
Question #2
0 Two planes leave Provo airport traveling in opposite
directions. One plane travels at 520mph and the
other at 480mph. How long will it take them to be
3000 miles apart?
0 HINT: In this case, though not specifically stated, we understand they’ve been traveling for
the same amount of time, 𝑡. And, since 3000 miles is a total distance for both planes
traveling in opposite directions, we can add the distance of each for the total distance, 𝐷 =
𝑟1 𝑡 + 𝑟2 𝑡.
Solution #2
D
r
t
Plane 1
D
520mph
𝑡
Plane 2
3000-D
480𝑚𝑝ℎ
𝑡
Total
3000
Plane 1: 𝐷1 = 520𝑡
Plane 2: 𝐷2 = 480𝑡
Since their times are equal, I could solve each for t and set them equal to
each other (like the previous problem), but since I actually want to solve
for the time and I know the total distance, I decide to add the 𝑟 ∙ 𝑡 of each
and set equal to the total distance
3000 = 𝐷1 + 𝐷2
3000 = 520𝑡 + 480𝑡
Question #3
0 Marie rode her bike from her house to her brother
Donny’s house, averaging 20mph. Later Donny drove
her home in his car at 30mph. If Marie’s total travel
time was 3 hours, how far away does she live from
Donny?
0 HINT: We understand that the distance in each direction is equal.
Solution #3
D
r
t
Bike
D
20mph
𝑡1
Car
D
30mph
𝑡2
3
0 𝐷 = 20𝑡1
0 𝐷 = 30𝑡2
0 Since we want the distance
between their houses and we
know the total time, I solve
each for 𝑡 and add the two
times together.
𝐷
𝐷
𝑡1 = 20 & 𝑡2 = 30
𝐷
𝐷
𝑡1 + 𝑡2 =
+
20 30
3=
𝐷
𝐷
+
20 30
𝐷 = 36
Question #4
0 If a hole is 4 feet in diameter, how deep must the hole be in order
to hold 113 cubic feet of dirt. Round to the nearest whole foot.
0 HINT: How do you measure the volume of a cylinder?
Solution #4
0 We know the volume of a cylinder is the area of a
circle multiplied by its height.
𝑉 = 𝜋𝑟 2 ℎ
And
𝑑 = 2𝑟 → 𝑟 = 2
So,
113 = 𝜋22 ℎ
ℎ ≈ 9 𝑓𝑡.
Question #5
0 The width of a rectangle is 3 less than one-half its
length, and the perimeter of the rectangle is 51 feet.
Find its dimensions.
0 HINT: This is a geometry problem that requires careful word translation. See English to Math
Translation handout.
Solution #5
0 We know for a rectangle:
𝑃 = 2𝑙 + 2𝑤
We are given
1
𝑤 = 𝑙−3
2
𝑃 = 51
And so by substitution,
1
51 = 2𝑙 + 2 𝑙 − 3
2
𝑙 = 19 𝑓𝑡.
𝑤 = 6.5 𝑓𝑡.
Question #6
0 The perimeter of a rectangle is 38 inches. The width
of a rectangle is 1 inch more than half its length. What
is the area of the rectangle?
Solution #6
Here we are dealing with two rectangular formulas:
𝑃 = 2𝑙 + 2𝑤
𝐴=𝑙∙𝑤
We are given
1
𝑤 = 𝑙+1
2
𝑃 = 38
By Substitution
1
38 = 2𝑙 + 2 𝑙 + 1
2
𝑙 = 12
𝑤=7
Then the 2nd part
𝐴 = 12 ∙ 7 = 84 𝑠𝑞. 𝑓𝑡.
Question #7
0 A trapezoid has an area of 35 square centimeters.
Assuming that one base is 9 centimeters and the other
base is 11 centimeters, find the height of the
trapezoid.
0 HINT: What is the formula for the area of a trapezoid?
Solution #7
0 The formula for a trapezoid is
1
𝐴 = ℎ 𝑏1 + 𝑏2
2
(See the next slide for a tip on how to remember this without “memorizing” it.)
So, plugging in the information
1
35 = ℎ 9 + 11
2
ℎ = 3.5 cm
Formula for a Trapezoid
0 Imagine if you cut a trapezoid in half from diagonal corners.
0 We get two triangles. We know the area of a triangle is
1
𝐴 = 𝑏ℎ
2
So now adding the area of the two triangles, each with a different base, but both
with the same height (which is always the case of a trapezoid):
1
1
𝐴 = 𝑏1 ℎ + 𝑏2 ℎ
2
2
And by factoring
1
𝐴 = ℎ(𝑏1 + 𝑏2 )
2
Question #8
0 You need a 15% acid solution for a certain test, but
you only have a 10% solution and a 30% solution. You
need 10 liters of the 15% acid solution. How many
liters of 10% solution and 30% solution should you
use?
Solution #8
1. Read the question.
0 need a 15% acid solution
0 have a 10% solution and a 30% solution
0 need 10 liters of the 15% acid solution
0 How many liters of 10% solution (A) and 30% solution (B) should you use?
2. Identify type of problem.
0 Mixture
3. Set up approach to solving. (Draw a picture to help.)
0 In this case we have 2 unknowns so will need to equations.
Amount of
Solution
Solution A
.10
=
×
.30
=
10 liters ×
.15
= .15(10)
𝐴 + 𝐵 = 10
. 10𝐴 + .30𝐵 = .15(10)
.10𝐴
+
.30𝐵
=
Mixture
Needed
×
𝐴
+
𝐵
=
Solution B
% of Acid
Solution #8 con’t
4. Do the math.
0 𝐴 = 7.5 liters
0 𝐵 = 2.5 liters
5. Read the final question; does the answer make
sense in the context of the problem? (i.e. distance
is never negative)
Question #9
0 Find the selling price per pound of a coffee mixture
made from 8 pounds of coffee that sells for $9.20 per
pound and 12 pounds of coffee that costs $5.50 per
pound.
Answer:
$6.98/lb
Solution #9
Weight in lbs.
Price per lb.
Total Cost
Coffee 1
8lbs
$9.20/lb
8(9.2)
Coffee 2
12lbs
$5.50/lb
12(5.5)
Coffee
Mixture
20lbs
𝑥
20𝑥
8 9.2 + 12 5.5 = 20𝑥
Question #10
0 How many liters of a 70% alcohol solution must be
added to 50 liters of a 40% alcohol solution to
produce a 50% alcohol solution?
0 HINT: The 50% alcohol solution will be a total liters of (𝑥 + 50).
Answer:
70% alcohol solution = 25 liters
Solution #10
# of liters
% of alcohol
Total
70% solution
𝑥
.7
.7𝑥
40% solution
50
.4
.4(50)
Mixture
𝑥 + 50
.5
.5(𝑥 + 50)
.7𝑥 + .4 50 = .5(𝑥 + 50)
Question #11
0 A mixture contains alcohol and water in the ratio 5:1.
After the addition of 5 liters of water, the ratio of
alcohol to water becomes 5:2. Find the quantity of
alcohol in the original mixture.
0 HINT: This sound like a mixture problem. However, we have only one container to which we
add water, so it’s not really a mixture problem and we don’t need a table. Because we see a
ratio, we will set this up as a ratio problem. Also, note that a ratio of 5:1 can also be
5
expressed as
1
Solution #11
0 Given
0 A mixture of alcohol to water in the ratio of 5:1.
0 This means for every 5 liters of alcohol, we have 1 liter of water.
𝑎
𝑤
0 We can rewrite a ratio, 𝑎: 𝑤 = . So,
0
0
0
0
𝑎 5
=
𝑤 1
We add 5 liters of water to the mixture and have a new ratio of 5:2.
𝑎
5
=
𝑤+5 2
By cross multiplication we have
𝑎 = 5𝑤
2𝑎 = 5𝑤 + 25
By substitution
2 5𝑤 = 5𝑤 + 25
𝑤=5
And solving for 𝑎
𝑎 = 5 5 = 25
Question #12
0 Helen Weller invested $14,000 in an account that pays
10% simple interest. How much additional money
must be invested in an account that pays 13% simple
interest so that the average return on the two
investments amounts to 11%?
Solution #12
P
Account 1
$14,000
Account 2
𝑥
Both
Accounts
$14,000 + 𝑥
r
t
10%
1
13%
1
11%
1
I
.1($14,000)
+.13𝑥
=.11($14,000+x)
.1($14,000) + .13𝑥 = .11($14,000 + 𝑥)
𝑥 = $7000
Why does time equal 1? This is because we are talking about a one time event, such as
after one year.
Question #13
0 If P dollars are invested at a simple interest rate r (in
decimals), the amount A that will be available after t
years is 𝐴 = 𝑃 + 𝑃𝑟𝑡. Determine the amount of
money that was invested if $1337.60 resulted from a
2-year investment at 10.8%.
0 HINT: We don’t really need a table here since we’re talking about only one account. Just plug
in the info.
Solution #13
Given:
𝐼 = 𝑃 = 𝑃𝑟𝑡, where
𝐼 = $1337.60, 𝑟 = 10.8% = .108, 𝑡 = 2
We have
$1337.60 = 𝑃 + 𝑃(.108)2
𝑃 = $1100
Question #14
0 Mr. Holland invested $4900, part at 6% interest and
the rest at 8%. If the yearly interest on each
investment is the same, how much interest does he
receive at the end of the year?
0 HINT: The yearly interest on each investment is the same for each amount invested. This
means we can solve for I and set the two equal to each other.
Solution #14
I
Account 1
Account 2
𝐼1
𝐼2
Total
Amount
P
r
P
6%
$4900-P
8%
t
Total
1
𝐼1 = 𝑃(.06)
1
𝐼2 =
(4900 − 𝑃)(.08)
$4900
Since we have
𝐼1 = 𝐼2
Then
. 06𝑃 = .08 4900 − .08𝑃
𝑃 = 2800
Question #15
0 President Obama can mow the White House lawn in 4
hours on his tractor mower. President Clinton mowed
the lawn in 5 hours on his. How long would it take
them to mow the lawn together.
0 HINT: Obama does ¼ of the job in one hour and Clinton does 1/5 of the job in one hour.
How much of the job could they do together in one hour?
Solution #15
0 Using out work – ratio problem set-up, we have
1 1 1
+ =
4 6 𝑥
𝑥 = 2.4 hours
Question #16
0 Juan has five times as many girlfriends as Pedro.
Carlos has one girlfriend less than Pedro. If the total
number of girlfriends between them is 20, how many
does each gigolo have?”
~ Jaime Escalante (Stand and Deliver)
See the solution here.