3.1 Further Problem Solving
Interest
Simple Interest:
Interest = principal × rate × time
I = Prt
Ex. You deposit $9,200 in a savings account, which has a rate
of 4%. Find the interest at the end of the first year.
I=?
P = 9,200
r = 0.04
t=1
I = Prt
I = 9200(0.04)(1)
I = 368
$368 in interest after 1 yr.
Ex. You invest $20,000 in two accounts paying
7% and 9% annual interest. If the total
interest earned for the year was $1550, how
much was invested at each rate?
Ex. You invest $20,000 in two accounts paying 7% and 9% annual
interest. If the total interest earned for the year was $1550, how
much was invested at each rate?
Principal
×
Rate
=
Interest
Ex. You invest $20,000 in two accounts paying 7% and 9% annual
interest. If the total interest earned for the year was $1550, how
much was invested at each rate?
Principal
×
Rate
=
Interest
7% investment
x
0.07
0.07x
9% investment
20,000 – x
0.09
0.09(20,000 – x)
0.07x + 0.09(20,000 – x) = 1550
7% acct. has x = $12,500
9% acct. has 20,000 – x
0.07x + 1800 – 0.09x = 1550
= 20,000 – 12,500
-0.02x + 1800 = 1550
= $7,500
-0.02x + 1800 – 1800 = 1550 – 1800
-0.02x = -250
-0.02x = -250
Ans: $12,500 at 7%
-0.02 -0.02
$7,500 at 9%
x = 12,500
Mixture
Mixture:
solution ∙ rate of concentration = ingredient
Check out page 187
Ex. A 70 milliliter solution of acid in water contains 60%
acid. How much acid is in the solution?
solution = 70
solution ∙ concentration = ingredient
concentration = 0.6
70(0.6) = ingredient
ingredient = ?
42 = ingredient
Ans: 42 ml of acid
Ex. How many ounces of a 15% solution must
be mixed with 4 ounces of a 20% solution to
make a 17% solution?
Ex. How many ounces of a 15% solution must be mixed with 4
ounces of a 20% solution to make a 17% solution?
solution
∙
concentration
=
ingredient
Ex. How many ounces of a 15% solution must be mixed with 4
ounces of a 20% solution to make a 17% solution?
solution
∙
concentration
=
ingredient
15% solution
x
0.15
0.15x
20% solution
4
0.2
0.2(4)
17% solution
x+4
0.17
0.17(x+ 4)
0.15x + 0.2(4) = 0.17(x + 4)
-0.02x = -0.12
0.15x + 0.8 = 0.17(x) + 0.17(4)
-0.02 -0.02
x=6
0.15x + 0.8 = 0.17x + 6.8
0.15x + 0.8 – 0.17x = 0.17x + 0.68 – 0.17x
-0.02x + 0.8 = 0.68
-0.02x + 0.8 – 0.8 = 0.68 – 0.8
-0.02x = -0.12
Ans: 6 oz of 15% soln.
Motion
Motion:
distance = rate ∙ time
Ex. New York City and Washington, D.C., are about 240 miles apart.
A car leaves New York City traveling toward Washington, D.C., at
55 miles per hour. At the same time, a bus leaves Washington,
D.C., bound for New York City at 45 miles per hour. How long will
it take before they meet?
rate
∙
time
=
distance
Ex. New York City and Washington, D.C., are about 240 miles apart. A car
leaves New York City traveling toward Washington, D.C., at 55 miles per
hour. At the same time, a bus leaves Washington, D.C., bound for New
York City at 45 miles per hour. How long will it take before they meet?
rate
∙
time
=
distance
car
55
t
55t
bus
45
t
45t
55t + 45t = 240
100t = 240
100t = 240
100 100
t = 2.4
2.4 hours = 2hr + 0.4(60min)
= 2hr 24 min
Ans: 2.4 hours or
2 hours and 24 minutes
Value
Value:
amount of ingredient ∙ cost per unit of ingredient = value of ingredient
Ex. Jeremy paid for his breakfast with 36 coins consisting of dimes
and quarters. If the bill was $4.50, then how many of each type
of coin did he use?
# coins
∙
value per coin
=
total value
Ex. Jeremy paid for his breakfast with 36 coins consisting of dimes and
quarters. If the bill was $4.50, then how many of each type of coin did
he use?
# coins
dimes
quarters
∙
value per coin
=
total value
x
0.10
0.10x
36 – x
0.25
0.25(36 – x)
0.10x + 0.25(36 – x) = 4.50
0.10x + 0.25(36) – 0.25(x) = 4.50
0.10x + 9 – 0.25x = 4.50
-0.15x + 9 = 4.50
-0.15x + 9 – 9 = 4.50 – 9
-0.15x = -4.50
-0.15x = -4.50
-0.15 -0.15
x = 30
# dimes = x = 30 dimes
# quarters = 36 – x
= 36 – 30
= 6 quarters
Ans: 30 dimes
6 quarters