Download Black 2 - Step Equations

Solving 2-Step Equations – Black Problems
Work the following problems. For those problems in which the answers are not integers,
express the answer either as mixed numbers or as decimals, rounded to the nearest
hundredth.
1.
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 x be the number of days
that have passed.
a. Write the definition of x . Then write two expressions, one representing how much
Phil has after x days and the other representing how much Ernest has after x 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.
2.
Plumber Problem. Nick O’ Time, the plumber, charges $30 per hour. His brother, Ivan,
the plumber’s helper, charges $20 per hour. Nick starts working on a job. Four hours
later, Ivan joins him and both work until the job is finished.
a. If Nick has been working for x hours, how long has Ivan been working?
b. Write expressions for the number of dollars Nick has earned and for the number of
dollars Ivan has earned after x hours.
c.
The total bill for the job is $470. Write an equation stating this fact, and solve it to
find out how long Nick worked.
d. How much of the $470 does each one get?
3.
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.
Station
100 km/h
t hours
:
70 km/h
t hours
:
Patrol’s distance
?
10 km
Truck’s distance
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.
4. 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.
Willie:
2.9 km/m
? min
Bank
:
Willie’s distance
Robin:
1.7 km/m
t min
?
:
Robin’s distance
a. 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.
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?
5. 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 x be the number of hours Lois has been driving. In terms of x , how far
has she gone?
c.
In terms of x , how many hours has Superman been flying? How far has he
flown in this number of hours?
d. Write an equation involving x that is true when Superman catches up with
Lois.
e. When does Superman catch up with Lois? How far are they from Metropolis
then?
6. 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
Interceptor Missile
1’s dist
?
:
:
M’s distance
USA
?
2800 km
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.
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?
7. For what value of n is the following equation true? Express your answer as a mixed
number.
3
4+
=1
n
3+
1
7
8. A cliff at Warloe Bay is 36 feet above the water. If an object is dropped from
the cliff, the formula h = 36 - 16t 2 will give the height above the water’s surface,
in feet, of the object at t seconds from the time it was dropped. How many
seconds does it take for an object dropped from the cliff to reach the water’s
surface? Express your answer as a decimal to the nearest tenth.
9. How many distinct solutions are there to the equation x - 7 = x + 1 ?
10. Solve for x: 2 - 4 = 1 . Express your answer as a common fraction.
5 3 x
11. Rachel subtracted two positive numbers and the difference was 12. Britt
multiplied the same two numbers and the product was 540. What is the sum of
the numbers?
12. Gligs and Crocs. You have a rectangle which is 6 gligs long and 2 gligs wide. The
perimeter of this rectangle is 16 gligs long and its area is 12 square gligs.
When measured in crocs the perimeter of the rectangle in crocs is numerically
equal to its area in square crocs. How many crocs in 1 glig?
13. Translate each sentence into an algebraic equation, then solve each equation,
show all working and complete a check.
a. The sum of ten and the square of a number all divided by the product of two
and the same number is equal to thirteen divided by the same number.
b. Six consecutive multiples of negative three add to give negative sixty three.
What is the average of the six consecutive numbers?
14. Solve the equation and show all working out.
6 m 5 + m (4 m 2 - 2) = 2 m 2 (3 m
3
+ 2m) + 2m
15. Dinner Date. It is 12:10 p.m. You are to be at your friend’s house for dinner at
6:30 p.m. You have been studying all morning, and feel the need to get some
exercise.
a. How far can you quick-jog at an average speed of 10.5 mph and still be back in
time for dinner, knowing that you will be walking back by the same route at
3.75 mph and will need to allow 3 of an hour to shower, dress and drive to
4
your friend’s house? Write and solve an equation to get your answer.
b. At what average speed, and how far can you travel in one direction, if you
maintain the same rate of speed jogging out and back?
16. A man makes a trip of 10 km at x km/hr followed by a trip of 20 km at a rate of 5
km/hr faster. What is the total time taken? On average how long did each part of
the trip take?
Solving 2-Step Equations – Black Solutions
1.
Money Problem.
a.
b.
c.
x = no. of days.
100 - 3 x = Phil’s no. of dollars.
20 + 5 x = Ernest no. of dollars.
i Phil has $24 more.
ii Ernest has $32 more.
After 10 days.
d. 100 - 3 x = 100 – 3(10) = 70
2.
same
20 + 5 x = 20 + 5(10) = 70.
Plumber Problem.
a.
x
x
= Nick’s no. of hours.
- 4 = Ivan’s no. of hours.
b. 30 x = Nick’s no. of dollars.
20( x - 4) = Ivan’s no. of dollars.
c.
11 hours.
d. Nick $330, Ivan $140.
3.
Truck and Patrol Car Problem.
a. t = no. of hr for patrol car. 100t = no. of km for patrol car.
b.
c.
10 + 70t = no of km for truck.
i Truck is 5 km further.
ii Patrol car is 5 km further.
1 hour or 20 minutes.
3
⎛1⎞
d. 100t = 100 ⎜ ⎟ = 33 1
3
⎝ 3⎠
same
⎛1⎞
⎟
⎝3⎠
10 + 70t = 10 + 70 ⎜
= 33
1.
3
4.
Pursuit Problem.
a. t = Robin’s no. of minutes. 1.7t = Robin’s no. of km.
b. t – 5 = Willie’s no. of minutes. 2.9(t – 5) = Willie’s no. of km
c.
5.
About 12.08 min after R. starts.
d. About 20.54 km from bank.
Lois and Superman Problem.
a. (diagram)
b.
x = Lois’ no. of hours
50 x = Lois’ no of km
c.
x - 3 = Superman’s no. of hours
300( x - 3) = Superman’s no. of km
d. 50 x = 300( x - 3)
e. After 3.6 hours and they are 180 km from Metropolis.
6.
Missile Problem.
a. 2800 – 300t
b.
c.
i 1000 km
ii 4000 km
431(t – 7)
d. 2800 – 300t = 431(t – 7);
t = 7.96 min; about 413 km from station
7.
An interesting way to solve this problem is with the “cover-up” method. That means that a term (or
terms) is covered that involve the unknown, and the value of the part that is covered is calculated.
First, cover the denominator. Ask, “Three divided by what number equals one?” Obviously, when the
denominator is equal to 3.
This then gives a new equation,
3
= 3. Cover the fraction which contains the variable, and
n
4+
3+
1
7
ask, “Four plus what number equals three?” The covered part, then, must equal -1, because 4 plus the
covered part must equal 3.
This gives yet another equation
n
3+
1
7
= -1. Cover the unknown, and ask, “ Three-and-one-seventh
divided by what number equals negative one?”
⎛
⎝
Obviously - ⎜ 3 +
8.
1⎞
⎟
7⎠
= -3
1.
7
When the object reaches the water it will be at height zero. We must solve 0 = 36 - 6t 2 for t. If we
subtract 16t 2 from both sides, we get 16t 2 = 36. Then we can divide both sides by 16, which gives us
t 2 = 36 . Taking the square root of both sides, we get t = 6 or 3 . This means it will take 1.5
16
4
2
seconds for an object dropped from the cliff to reach the water’s surface.
(Note: Although -1.5 is also a solution to the equation, a negative result does not make sense here.)
9.
For each absolute value, we must consider both a positive a negative value. For this reason, we might
expect four solutions. If x - 7 and x + 1 are both positive before the absolute value is taken, then
we would have the equation x - 7 = x + 1, for which there is no solution. If they are both negative,
we would have –( x - 7) = -( x + 1), for which there is again no solution. The only way this can work is
if one expression is positive and the other is negative. This gives – ( x - 7) = x + 1 or x - 7 = - ( x +
1). Solving the first if these, we get - x + 7 = x + 1. Adding x to both sides and subtracting 1 from
both sides, we get 6 = 2 x , for which the solution is x = 3. Solving the second possibility, we get x
- 7 = - x - 1. Adding x to both sides and adding 7 to both sides, we get 2 x = 6, for which the
solution is again x = 3. In the end, there is only 1 solution. An alternate solution is to use a graphing
calculator. Entering and graphing the equations y =
x -7
and y =
of intersection to be (3,4). Therefore, the equations are equal only at
10.
x +1
x
will show the only point
= 3.
First we should do the subtraction on the left, using the least common denominator of 15:
1
6
x ; 15 -
20 = 1 ; - 14 = 1 . If the reciprocal of x is - 14 , then x must be - 15 .
14
15 x 15 x
15
2- 4 =
5 3
11.
Two numbers whose difference is 12 and whose product is 540 are needed. Look for factors of 540
that differ by 12. Trail and error finds 18 and 30: 30 + 18 = 48.
12.
Gligs and Crocs.
Let L be the length of the rectangle in gligs and W the width in gligs. The perimeter is then P = 2(L +
W) gligs and the area is A = L • W square gligs. If x equals the number of crocs in a glig then
the perimeter in crocs is 2 x (L + W) or 2 x L + 2 x W crocs, and the area in crocs is xL • xW or
x 2LW square crocs. Substituting the given values, Pcroc = 2 x (6 + 2) = 16 x and Acroc = x 2 • 6 • 2
4 , so there are
3
4 crocs in a glig. Therefore the length of the given rectangle is 8 crocs, and the width is 8 crocs.
3
3
= 12 x 2. Since we require that Pcroc =
Acroc numerically, then 16 x
⎛
⎝
The perimeter of the rectangle is 2 ⎜ 8 +
⎛8⎞
⎜ ⎟
⎝3⎠
13.
a.
or
8⎞
⎟
3⎠
or
= 12 x 2 and
10 + x 2 = 13
2x
x
x 3 = 26 x
x 3 = 16 x
x 2 = 16
x
= 4 or -4
b. -3 x + -3( x + 1) + -3( x + 2) + -3( x + 3) + -3( x + 4) + -3( x + 5) = -63
-18 x - 45 = -63
=
64 crocs, and the area of the rectangle is 8 •
3
64 square crocs.
3
10 x +
x
-18 x = -18
x =1
Therefore, the integers are -3, -6, -9, -12, -15, -18
The average of these 6 numbers is -10.5.
A shortcut to this problem would be to recognize that the average of 6 numbers is equal to
their sum divided by 6. Therefore, since the sum was given as -63, the average must be -63 ' 6
= -10.5.
14.
6m 5 + 4m
3
- 2m = 6 m 5 + 4 m
3
+ 2m
-2m = 2m
0 = 4m
4
4
0 = m
15.
Dinner Date.
The person has 335 minutes, i.e. 5.58 hours, to exercise (6 hours and 20 minutes = 380 minutes,
from which 45 minutes must be subtracted).
a. Since the distance out and back will be the same, the equation 10.5 x = 3.75(5.58 - x ) can be
utilized; therefore
allocated time.
x
= 1.47 hours, and you could cover 15.4 miles in each direction within the
b. The second part of this question requires the realization that since the time allocation in each
5.58 or 2.79 hours), setting any rate will determine the distance, and
direction is fixed (
2
setting any distance will determine the rate.
16.
The total time taken equals the time taken for the first part plus the time taken for the second
part. Let x = rate on initial leg of trip.
The total time equals the sum of the two times for each part of the trip (t 1 and t 2 ) where t 1 =
10 + 20 .
x x +5
1
5+
The average of these two times is (t 1 + t 2 ) =
2
x
and t 2 =
20
x +5
. This means t 1 + t 2 =
10
x +5
.
10
x
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
1-6
Algebra I: Expressions, Equations, and
Applications (Hardcover)~ Paul A.
Foerster, Addison-Wesley Publishing
Company, Menlo Park, CA, 1999
7 - 11
Math Counts (http://mathcounts.org)
12 , 15
http://balancedassessment.concord.org/
16
Cook, Allen, and Natalia Romalis.
Content Area Mathematics for
Secondary Teachers The Problem
Solver. New York: Christopher-Gordon,
Inc., 2006.