Transcript 3.5 Linear Equations and Problem Solving Word Problems

Linear Equations and Problem Solving
Word Problems!!!
Keys to succeed!
Write down important information
Put the info in a chart if you can
Consecutive Integers
Find three consecutive integers whose sum is
162.
Integer 1
Integer 2
Integer 3
Total
x-1
x
x+1
162
x  1  x  x  1  162
3 x  162
x  54
x  1  54  1  53
x  1  54  1  55
Traveling
A pair of hikers, 18 miles apart, begin at the same time to
hike toward each other. If one walks at a rate that is 1 mph
faster than the other, and if they meet 2 hours later, how
fast is each one walking?
Hiker 2’s dist.
Hiker 1’s dist.
18
Hiker 1’s distance + Hiker 2’s distance = 18
Hiker 1
x
2
2x
Hiker 2
x+1
2
2(x+1)
Rate
Time
Distance = 18
2 x  2( x  1)  18
2 x  2 x  2  18
4 x  2  18
4 x  16
x  4 mph
x  1  4  1  5 mph
Traveling
A pair of cars, 280 miles apart, begin at the same time to run
toward each other. If car A from city A runs at a rate that is 10
mph faster than car B from city B, and if they meet 2 hours later,
how far is the place they meet away from city A?
Car A’s dist.
Car B’s dist.
A
B
280
Car A’s distance + Car B’s distance = 280
Car A
x + 10
2
2(x + 10)
Car B
x
2
2x
Rate
Time
Distance = 280
2 x  2( x  10)  280
2 x  2 x  20  280
4 x  20  280
4 x  260 2( x  10)  2 75  150 mi.
x  65 mph
x  10  75 mph
Traveling
The Yankee Clipper leaves the pier at 9:00am at 8 knots (nautical
miles per hour). A half hour later, The Riverboat Rover leaves
the same pier in the same direction traveling at 10 knots. At
what time will The Riverboat Rover overtake The Yankee
Clipper?
Yankee
Clipper
9:00 ~ 9:30
Traveled
4 nt. miles
8
x hours after
9:30
8x
8x + 4
Riverboat
Rover
9:00 ~ 9:30
Traveled
0 nt. miles
10
x hours after
9:30
10x
0 + 10x
rate
time
dist.
total
Yankee Total = Riverboat Rover Total
4  8 x  0  10 x
4  2x
x  2 hr.
4 nt. mi.
YC
RR
9:00
8x nt. mi.
YC
10x nt. mi.
9:30
YC
RR
x hr. after
9:30
Tickets
The school play sold 450 tickets for a total of $1160. If
student tickets are $2.00 and adult tickets are $4.00,
how many of each type were sold?
Student
Adult
Total
2
4
-----
x
450 – x
450
2x
4(450 – x)
1160
Student tickets sales + Adult tickets sales = 1160
2 x  4(450  x)  1160
2 x  1800  4 x  1160
2 x  1800  1160
2 x  640
x  320 tks
450  x  450  320  130 tks
Tickets
Fred is selling tickets for his home movies. Tickets for friends are $3.00 and everyone else must pay
$5.00 per ticket. If he sold 72 tickets and made $258 how many of each type did he sell?
Friend
Non-Friend
Total
3
5
----
3x  5(72  x)  258
3x  360  5 x  258
2 x  360  258
2 x  102
x
72 – x
72
3x
5(72 – x)
258
x  51 tks
72  x  72  51  21 tks
Accounting
Barney has $450 and spends $3 each week. Betty has
$120 and saves $8 each week. How many weeks will it
take for them to have the same amount of money?
Barney
Betty
450
120
initial
3
8
wk spend
450  3x  120  8 x
450  120  11x
330  11x
x  3 wk
x
x
wk
450 – 3x
120 + 8x
end total
Accounting
You Try This!
Fred has $100 and saves $4 each week. Wilma has $28 and
saves $10 each week. How long will it take for them to have the
same amount of money? What is that amount?
Fred
Wilma
100
4
x
28
10
x
initial wk sp wk
100  4 x  28  10 x
100  28  6x
72  6x
x  12 wk
100 + 4x
28 + 10x
end total
28  10 x  28  10 12  $148
More on Traveling
A driver averaged 50mph on the highway and 30mph
on the side roads. If the trip of 185 miles took a total
of 4 hours and 30 minutes, how many miles were on
the highway.
Highway
Side Road
50
30
Total
x 185  x

 4.5
50
30
x
185 – x
185
x/50
(185 – x)/30
4.5
My God! It is so complicated!!!
More on Consecutive Integers
Find three consecutive integers that the difference of
the product of two larger ones and the product of two
smaller ones is 30.
Integer 1
Integer 2
Integer 3
Prod. of Larger 2
Prod. of Smaller 2
x( x  1)  x( x  1)  30
x 2  x  x 2  x  30
2 x  30
x-1
x
x+1
x(x + 1)
x(x - 1)
x  15
x  1  16
x  1  14
More on Traveling
A driver averaged 50mph on the highway and 30mph on the side
roads. If the trip of 185 miles took a total of 4 hours and 30
minutes, how many miles were on the highway.
Highway
Side Road
50
30
Total
50 x  30(4.5  x)  185
50 x  135  30 x  185
20 x  135  185
x
4.5 – x
4.5
50x
30(4.5 – x)
185
20 x  50
x  2.5 hr.
50 x  50(2.5)  125 mi.
Weighted Averages
You have 32 coins made up of dimes and nickels.
You have a total of $2.85. How many of each
type of coin do you have?
Dime
Nickel
Total
10
5
x
32 – x
32
10 x  5(32  x)  285
10 x  160  5 x  285
5 x  160  285
10x
5(32 – x)
285
5 x  125
x  25
32  x  7
Weighted Averages
The Quick Mart has two kinds of nuts. Pecans
sell for $1.55 per pound and walnuts sell for
$1.95 per pound. How many pounds of walnuts
must be added to 15 pounds of pecans to make
a mixture that sells for $1.75 per pound.
Pecans
Walnuts
Mixture
1.55
15
15 · 1.55
1.95
x
1.95x
1.75 x +15 1.75(x + 15)
1.55  15  1.95x  1.75( x  15)
23.25  1.95 x  1.75 x  26.25
23.25  0.2 x  26.25
0.2 x  3
x  15 lb.
Mixture
A druggist must make 20 oz of a 12% saline
solution from his supply of 5% and 15% solutions.
How much of each should he use?
12%
solution
12%
20
20·12%
5%
solution
5%
x
x · 5%
15%
solution
15%
20 – x
(20 – x) ·15%
20  0.12  0.05x  (20  x)  0.15
2.4  0.05 x  3  0.15 x
x  6 oz.
2.4  0.1x  3
20  x  20  6  14 oz.
0.6  0.1x