2.5 An Introduction to Problem
Solving
Steps
• See table on page 148 and steps on page 149
Number Problem
Ex. The sum of five and three times a number is
59. Find the number.
Ex. The sum of five and three times a number is
59. Find the number.
x = the #
5 + 3x = 59
5 + 3x – 5 = 59 – 5
3x = 54
3x = 54
3 3
x = 18
the number is 18
Two Unknowns
Ex. Page 156, #21: Time spent watching TV
exceeded time spent listening to the radio by
581 hours. The combined time devoted to
these two media was 2529 hours. In 2007,
how many hours did Americans spend
listening to radio and how many hours were
spent watching TV? (Hint on page 151)
Two Unknowns
Ex. Page 156, #21: Time spent watching TV exceeded time spent listening to
the radio by 581 hours. The combined time devoted to these two media
was 2529 hours. In 2007, how many hours did Americans spend listening
to radio and how many hours were spent watching TV? (Hint on page 151)
x = time spent listening to radio
x + 581 = time spent watching TV
x + (x + 581) = 2529
2x + 581 = 2529
2x + 581 – 581 = 2529 – 581
2x = 1948
2
2
x = 974
time spent on radio = 974 hours
time spend on TV = 974+581
=1,555 hours
Consecutive Integers
0
1
2
x
x+1
3
4
5
6
1st consecutive integer = x
2nd consecutive integer = x + 1
Consecutive Even Integers
0
1
2
x
3
4
5
6
x+2
1st consecutive even integer = x
2nd consecutive even integer = x + 2
Consecutive Odd Integers
0
1
x
2
3
4
5
6
x+2
1st consecutive odd integer = x
2nd consecutive odd integer = x + 2
Consecutive Integers
Ex. Page 156, #28: Babe Ruth and Hank
Greenberg are among the ten baseball players
with the most home runs in a major league
season. The number of home runs by Ruth in
1927 and Greenberg in 1938 are consecutive
even integers whose sum is 118. Determine
the number of homers hit by Greenberg and
by Ruth.
Ex. Page 156, #28: Babe Ruth and Hank Greenberg are among the ten
baseball players with the most home runs in a major league season.
The number of home runs by Ruth in 1927 and Greenberg in 1938
are consecutive even integers whose sum is 118. Determine the
number of homers hit by Greenberg and by Ruth.
x = # home runs hit by Greenberg
x + 2 = # home runs hit by Ruth
x + (x + 2) = 118
2x + 2 = 118
2x + 2 – 2 = 118 – 2
2x = 116
2x = 116
2 2
x = 58
Greenberg hit 58 home runs
Ruth hit 58 +2 or 60 home runs
Base Price Plus Extra “per” Rate
Ex. Page, #31: In 2004, mortgage payments
averaged $960 per month. From 1975 to
2004, monthly mortgage payments increased
by approximately $11.40 per year. If this trend
continues, how many years after 2004 will
mortgage payments average $1074? In which
year will that occur?
Ex. Page, #31: In 2004, mortgage payments averaged $960 per
month. From 1975 to 2004, monthly mortgage payments
increased by approximately $11.40 per year. If this trend
continues, how many years after 2004 will mortgage
payments average $1074? In which year will that occur?
x = # years after 2004
mortgage + increase per year = 1074
960 + 11.40x
= 1074
960 + 11.40x – 960 = 1074 – 960
11.4x = 114
11.4x = 114
11.4 11.4
x = 10 10 years after 2004, so
in 2014
Geometry
Ex. An American football field is a rectangle
with a perimeter of 1040 feet. The length is
200 feet more than the width. Find the width
and length of the rectangular field.
Ex. An American football field is a rectangle with a perimeter of 1040
feet. The length is 200 feet more than the width. Find the width
and length of the rectangular field.
x = width
x + 200 = length
P = 2L + 2W
1040 = 2(x + 200) + 2(x)
1040 = 2x + 400 + 2x
1040 = 4x + 400
1040 – 400 = 4x + 400 – 400
640 = 4x
640 = 4x
4
4
160 = x
x
x + 200
width = x = 160 feet
length = x + 200
= 160 + 200
= 360 feet
Reduction/Increase
Ex. After a 20% reduction, you purchase a
television for $320. What was the television’s
price before the reduction?
Ex. After a 20% reduction, you purchase a television for $320.
What was the television’s price before the reduction?
x = original price of TV
original price – reduction = new price
original price – 20%reduction = new price
x –
0.2x
= 320
1x – 0.2x = 320
0.8x = 320
0.8x = 320
0.8 0.8
x = 400 orig price = x = $400