Problem Solving with Equations – Blue Problems
Solving Multi-Step Equations with Variables on Both Sides
Solve. Check your solution.
1. 4(7 - 2 ) – 6(
2.
+
3.
+1=5
4.
- 1) = 10(
+ 5)
= 21
-
5.
=1
=4
6.
= -5
7.
=
8.
=a
-
9. a -
+
-
+
-
=0
10. Set up an equation, and solve it to answer the following:
The sum of three numbers is 156. The second number is 5 more than the
first number, and the third number is 5 times the second number.
What are the numbers?
11. Tom’s mother was 28 years old when Tom was born. She is now three times
his age. How old are they both now?
12. Frank is 23 years older than Ernest. In six years’ time, Frank will be twice
as old as Ernest will be. How old are they now?
13. A man is twice as old as his son. Nine years ago the sum of their ages was
66 years.
How old are they both now?
14. Laura has done five tests, each marked out of 20, and will be doing one
more test. Her average mark is 15.4. Her mother promised her a reward if
her average mark over the six tests is at least 16. What is the lowest mark
she can get in the sixth test so that she gets the reward?
15. The length of a rectangle is 3 more than twice its breadth. Its perimeter is
48 meters. What is its area?
16. I have two jugs. One holds 500 milliliters more than the other. The larger
jug is half full of water. When I pour all this water into the smaller jug, it
becomes two-thirds full. How many milliliters do my jugs hold?
17. Katherine is 8 years younger than Eryn. Peter is two-thirds the age of
Katherine. Fred is 15 years older than Eryn. Their combined ages add up to
130 years.
How old is Katherine?
18. What integer, added to both the numerator and the denominator of
,
results in a fraction with a value of 0.8?
19.
Lemonade Fun. Jose and Nick decide to sell lemonade on
a hot summer day. They’re selling two sizes: a 25-cent cup
and a smaller 10-cent cup. At the end of the day, they’ve
made $8.45.
They count the number of empty cups remaining. They started out with
equal numbers of large and small cups, but now have three more small cups
than large. How many cups of each size lemonade did they sell?
For Exercises 20 – 22, solve the equation.
20. 7 +
21.
y=
(r + 1) =
y-5
(r + 14)
22.
Example:
(4
=
(3
- 6)
+3=
Solution:
Multiply each side by
=
+3
23. 6 -
- 5) -
+3
=7
Distribute the
-
=7-
Subtract
3
=
Simplify.
=
Divide each side by 3.
24.
=
29. Consider the equation 12
equation have a solution?
+ a = 4(3
.
.
from each side.
+5=-
- 8). For what value(s) of a does the
30. Consider the two figures below. For what value of
are the areas of the
figures the same? The figures are composed of triangles and rectangles.
Example
Worm and Snail Problem. A worm starts at the oak tree and moves away,
heading for the elm tree at a constant rate of 13 meters per hour (m/h). At
the same time, a snail starts at the elm tree and moves toward the oak tree
at a constant rate of 17 m/h. The two trees are 100 m apart. Let
be the
number of hours the two creatures have been creeping.
a.
b.
c.
d.
e.
f.
__
Draw a diagram showing the two trees 100 m apart and the worm and
snail somewhere between the trees. Draw arrows marking each
creature’s distance from the oak tree.
Write the definition of . Then write an expression for each one’s
distance from the oak tree.
Who is closer to the oak tree after 2 hours? How much closer?
Who is closer to the oak tree after 4 hours? How much closer?
When do they pass each other?
How far are they from the oak tree when they pass each other?
___
_____
a.
(See Note 1.)
_ _ _ _ _
b.
13
_ _ _ _ _
= number of hours they have been going.
= number of meters worm is from oak tree.
(See Note
2.)
100 - 17
= number of meters snail is from oak tree.
_____
(See Note 3.)
_____
c. Worm:
13
= 13(2)
= 26
Worm is closer
Snail:
100 - 17
= 100 – 17(2)
= 66
by 40 m.
d. Worm:
13
= 13(4)
= 52
Snail:
100 - 17
= 100 – 17(4)
= 32
Snail is closer by 20 m.
_____
_____
Think These Reasons
e. 13
30
= 100 - 17
Let the two distances be equal. (See Note 4.)
= 100
Add 17
=
Divide each member by 30.
hours
Answer the question.
_____
_____
f. Worm:
13
or
Snail:
100 - 17
= 13
= 100 - 17
= 13
= 100 -
=
= 100 - 56
= 43
= 43
43
to each member.
meters
43
meters
(See Note 5.)
Notes:
1. Show enough information in the diagram to show clearly which distance
is which.
2. 13 is the rate and
is the time; distance = rate x time.
3. The snail has gone a distance of 17 m (rate x time). So its position is
100 - 17 meters from the oak tree.
4. When they pass each other, each one is the same distance from the oak
tree. So you let their distances equal each other.
5. You can find either the worm’s distance or the snail’s distance, since
both are equal. Finding both distances gives you a check on the
correctness of your answer.
31.
Money Problem. Phil T. Rich has $100 and spends $3.00 of it per day.
Ernest Worker has only $20 but is adding to it at the rate of $5.00 per
day. Let
be the number of days that have passed.
a.
Write the definition of . Then write two expressions, one representing
how much Phil has after
days and the other representing how much
Ernest has
days.
b. Who has more money, and how much more, after:
i 1 week;
ii 2 weeks?
c.
After how many days will each have the same amount of money?
Write and solve an equation to find this number of days.
d. Show that each actually does have the same amount of money after the
number of days you calculated in part c.
32.
Weight Change Problem. Fred weighs 187 pounds but is on a diet that
makes him lose 1.7 pounds per week. Joe weighs only 93 pounds but is on a
diet that makes him gain 0.9 pounds per week.
a.
Write an expression representing Fred’s weight after
weeks and
another expression representing Joe’s weight after
weeks.
b. What is each one’s weight after:
i 10 weeks;
ii one year?
c.
33.
After how many weeks will each be the same weight? Show your work.
Waitress and Cook Problem. The waitress at the Greasy Spoon Café makes
wages of $32 per day and the cook makes $50 per day. In addition, they
divide the tips received in such a way that the waitress gets 70% of the tip
money and the cook gets 30% of it. (“70% of ….” means “0.7 times ….”) Let
be the number of dollar tips in a day.
a.
Write an expression representing the total amount of money (wages
plus tips) the waitress makes and the total amount the cook makes a
day.
b. How much does each make if there are:
i $20 in tips;
ii $100 in tips?
c.
34.
What amount of tip money makes each person get the same total
number of dollars per day?
Bathtub Problem. Suppose that you turn on the hot water, which flows at
8.7 liters per minute into the bathtub. Two minutes later you also turn on
the cold water, which flows at 13.2 liters per minute. Let
be the number
of minutes since you turned on the cold water.
a.
Write expressions in terms of
for the number of minutes the hot
water has been running, the number of liters the hot faucet has
delivered, and the number of liters the cold faucet has delivered.
b. Write an equation stating that the hot and cold faucets have delivered
the same number of liters. Solve the equation to find out when this
happens.
c.
35.
The tub holds 100 liters. Will it have overflowed by the time the hot
and cold faucets have delivered the same amounts? Justify your
answer.
Truck and Patrol Car Problem. A truck passes a highway patrol station
going 70 kilometers per hour (km/h). When the truck is 10 kilometers past
the station, a patrol car starts after it, going 100 km/h. Let t be the
number of hours the patrol car has been going.
a.
Write the definition of t. Then write two expressions, one representing
the patrol car’s distance from the station and the other representing
the truck’s distance from the station after t hours.
b. If they continue at the same speeds, who will be farther from the
station, and how many kilometers farther, after:
i 10 minutes;
ii 30 minutes?
c.
At what time t does the patrol car reach the truck?
d. Show that the two distances really are the same at the time you
calculated in part c.
36.
Pursuit Problem. Robin Banks robs a bank and takes off in his getaway car
at 1.7 kilometers per minute (km/m). 5 minutes later Willie Katchup leaves
the bank and chases Robin at 2.9 km/m. Let t be the number of minutes
Robin has been driving.
Write the definition of t. Then write an expression representing Robin’s
distance from the bank in terms of t.
b. In terms of t, how long has Willie been driving? Write an expression
representing Willie’s distance from the bank in terms of t.
a.
c.
When Willie catches up with Robin, their distances from the bank are
equal. Write an equation stating this fact and solve it to find out when
Willie Katchup will catch up with Robin Banks.
d. Where does Willie catch Robin?
37.
Lois and Superman Problem. Lois Lane leaves Metropolis driving 50 km/h.
Three hours later Superman leaves Metropolis to catch her, flying 300
km/h.
a.
Draw a diagram showing Lois’s distance from Metropolis, Superman’s
distance from Metropolis, and the distance between them.
b. Let
be the number of hours Lois has been driving. In terms of
far has she gone?
, how
c.
In terms of , how many hours has Superman been flying? How far has
he flown in this number of hours?
d. Write an equation involving
with Lois.
that is true when Superman catches up
e. When does Superman catch up with Lois? How far are they from
Metropolis then?
38.
Missile Problem. A missile tracking station in the Pacific Ocean detects a
ballistic missile coming straight toward it at 300 kilometers per minute
from a test site in the continental United States.
Station
a.
Let t be the number of minutes since the missile was detected. At time
t = 0, the missile was 2800 kilometers from the tracking station. Write
an expression in terms of t for the missile’s distance from the station.
b. Where is the missile:
i 6 minutes after it is detected;
ii 4 minutes before it is detected?
c.
At time t = 7, the tracking station fires an interceptor directly toward
the oncoming missile. The interceptor travels at 431 kilometers per
minute. Write an expression for the interceptor’s distance from the
tracking station at time t. Think carefully about how many minutes the
interceptor has been traveling!
d. Write an equation that is true when the interceptor meets the missile.
At what value of t will they meet? How far from the tracking station
will they be?
Problem Solving with Equations – Blue Solutions
Solving Multi-Step Equations with Variables on Both Sides
1.
=
= 36
2.
3.
=
4.
=
=1
5.
6.
7.
8.
9.
=
m=
a=
a=1
10.
The numbers are 18, 23 and 115.
11.
Tom is 14 years old and his mother is 42 years old.
12.
Frank is 40 and Ernest is 17 years old.
13.
Their ages are 28 and 56 years.
14.
Laura needs at least 19 marks.
15.
The area is 119
16.
The jugs hold 1500 mL and 2000 mL.
17.
Katherine is 27 years old.
18.
19.
.
Solving this problem algebraically, we can set up
=
=
. Using the first and last
ratios, and cross-multiplying, we get the equation 4(7 + x) = 5(6 + x) or 4x + 28 = 5x + 30.
Subtracting 4x and 30 from each side yields -2 = x. By plugging the -2 into the proportion
above, we can see that this number does work.
Lemonade Fun.
Let L be the number of large cups that they sold. Since they were left with three more small
cups than large cups, they sold L-3 small cups.
The total amount of money made selling large cups was .25
selling small cups was .10
L and the total amount made
(L-3). Since these two amounts add up to $8.45, we know that:
.25L + .10(L-3) = 8.45
.25L + .10L – .30 = 8.45
.35L = 8.75
L=
= 25
L – 3 = 22
So they sold 25 large (25 cent) cups and 22 small (10 cent) cups.
20.
-24
21.
22.
Identity
23.
24.
29.
-32
30.
3
31.
Money Problem.
a.
b.
c.
= no. of days.
100 - 3 = Phil’s no. of dollars
20 + 5 = Ernest’s no. of dollars
i Phil has $24 more.
ii Ernest has $32 more.
After 10 days.
d. 100 - 3
32.
a.
= 100 – 3(10) = 70
Fred: w = 187 – 1.7x
same
Joe: w = 93 + 0.9x
20 + 5
= 20 + 5(10) = 70
b.
10 weeks; Fred = 170 , Joe = 102
One year; Fred = 98.6 , Joe = 139.8
c.
33.
, 36.2 weeks
a.
Waitress: Money = 32 + 0.7x
Cook: Money = 50 + 0.3x
b.
$20 in tips; waitress = $46 , cook = $56
$100 in tips; waitress = $102 , cook = $80
c.
34.
, x = $45
a.
(x+2) = minutes hot water has been running
8.7(x+2) = liters of hot water
13.2x = liters of cold water
b.
8.7(x+2) = 13.2x , x = 3.87 minutes
c.
35.
liters of water. The tub will overflow.
Truck and Patrol Car Problem.
a.
b.
c.
t = no. of hr for patrol car.
100t = no. of km for patrol car.
10 + 70t = no. of km for truck.
i Truck is 5 km further.
ii Patrol car is 5 km further
hour or 20 minutes.
d. 100t = 100
36.
= 33
Pursuit Problem.
a.
t = Robin’s no. of minutes.
1.7t = Robin’s no. of km.
same
10 + 70t = 10 + 70
= 33
.
b. t – 5 = Willie’s no. of minutes.
2.9(t – 5) = Willie’s no. of km.
c.
37.
About 12.08 min after R. starts.
d. About 20.54 km from bank.
Lois and Superman Problem.
a.
(diagram)
b.
= Lois’ no. of hours. 50
c.
- 3 = Superman’s no. of hours. 300(
d. 50
= 300(
= Lois’ no. of km.
- 3) = Superman’s no. of km.
- 3)
e. After 3.6 hours.
38.
Missile Problem.
a.
b.
c.
2800 – 300t
i 1000 km
ii 4000 km
431(t – 7)
d. 2800 – 300t = 431(t – 7); t = 7.96 min; about 413 km from station
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some
cases, sources were not known.
Problems
Bibliography Information
19
The Math Forum @ Drexel
(http://mathforum.org/)
18
Math Counts
(http://mathcounts.org)
1 - 17
Thomas, Alice, and Margaret
Jordan. Maths in the Fast
Lane. Putney: Phoenix
Education, 1998. Print
20 – 28, 39 - 42
Larson, Ron, Laurie Boswell,
Timothy D. Kanold, and Lee
Stiff. Algebra 1 Concepts and
Skills. Evanston: McDougal
Littell, 2001. Print.
31 - 38
Algebra I: Expressions,
Equations, and Applications
(Hardcover)~ Paul A.
Foerster, Addison-Wesley
Publishing Company, Menlo
Park, CA, 1999