[word problems part 4][section 29]

(1.) Sam has 45 coins in dimes and quarters. The total value

of the coins is $7.65 . How many coins of each kind

does he have?

d + q = 45

.10d + .25q = 7.65 here is the problem

- .10d - .10q = -4.5 multiply equation 1 thru by -.10
______________________
.15q = 3.15 subtract equations
_____ ______
.15 .15 divide each side by .15

q = 21 divide and cancel

d + 21 = 45 replace q with 21

-21 -21 subtract 21 from each side
_______________
d = 24 subtract

result: 24 dimes and 21 quarters

(2.) Cliff puts quarters and nickels aside for paying tolls.

He has a total of 18 coins with a total value of $2.90 .

How many coins of each kind does he have?

n + q = 18

.05n + .25q = 2.90 here is the problem

-.05n - .05q = -.90 multiply equation 1 thru by -.05
______________________
.2q = 2 subtract equations
___ ___
.2 .2 divide each side by .2

q = 10 divide and cancel

n + 10 = 18 replace q with 10

-10 -10 subtract 10 from each side
____________________
n = 8 subtract

result : 8 nickels and 10 quarters

(3.) Joe invested $5000, part at a yearly rate of 7% and

part at a yearly rate of 9%. The total interest

earned for one year was $368 . How much did he

invest at each rate?

x + y = 5000

.07x + .09y = 368 here is the problem

-.07x - .07y = -350 multiply eq 1 thru by -.07
_________________________
.02y = 18 subtract equations
_____ ____
.02 .02 divide each side by .02

y = 900 divide and cancel

x + 900 = 5000 replace y with 900

-900 -900 subtract 900 from each side
__________________
x = 4100 subtract

result: $4100 at 7 % and $900 at 9 % .

(4.) Two loans were taken out totaling $35,000. The yearly

interest rates were 12% and 15% and the total yearly

interest was $4650. Find the amount of each loan.

x + y = 35,000

.12x + .15y = 4650 here is the problem

-.12x - .12y = -4200 multiply equation 1 thru by -.12
_____________________________
.03y = 450 subtract equations
____ _____
.03 .03 divide each side by .03

y = 15,000 divide and cancel

x + 15,000 = 35,000 replace y with 15,000

-15,000 -15,000 subtract 15,000 fr ea side
____________________________
x = 20,000 subtract

result: $20,000 at 12 % and 15,000 at 15 % .

(5.) 7500 tickets were sold for a concert. Total receipts

amounted to $45,000 . Tickets sold for $5.50 and $7.00 .

How many tickets of each kind were sold?

x + y = 7500

5.5x + 7y = 45,000 here is the problem

-5.5x - 5.5y = -41,250 multiply eq 1 thru by -5.5
__________________________
1.5y = 3750 subtract equations
____ ____
1.5 1.5 divide each side by 1.5

y = 2500 divide and cancel

x + 2500 = 7500 replace y with 2500

-2500 -2500 subtract 2500 from each side
________________________
x = 5000 subtract

result: 5000 tickets at $5.50 per ticket,

2500 tickets at $7.00 per ticket

(6.) Five full-fare bus tickets and one half-fare ticket

cost $90.75 . Four full-fare tickets and two half-fare

tickets cost 82.50 . How much does each type of

ticket cost?

5f + h = 90.75

4f + 2h = 82.50 here is the problem

-2f - h = -41.25 multiply eq 2 thru by -1/2, cancel

5f + h = 90.75 put this here
____________________
3f = 49.50 add equations
___ _______
3 3 divide each side by 3

f = 16.50 divide and cancel

4(16.50) + 2h = 82.50 replace f with 16.50

66 + 2h = 82.50 multiply

- 66 - 66 subtract 66 from each side
________________________
2h = 16.50 subtract
___ ______
2 2 divide each side by 2

h = 8.25 divide and cancel

result: full-fare tickets cost $16.50 and half-fare

tickets cost $8.25 .

(7.) Bob makes a bank deposit of $595 with 96 five- and

ten- dollar bills. How many of each kind of bill

did she deposit?

f + t = 96

5f + 10t = 595 here is the problem

-5f - 5t = -480 multiply eq. 1 thru by -5
_____________________
5t = 115 subtract equations
___ ____
5 5 divide each side by 5

t = 23 divide and cancel

f + 23 = 96 replace t with 23

-23 -23 subtract 23 from each side
___________________
f = 73 subtract

result: 73 fives and 23 tens

(8.) Ralph paid $6.60 for some 15 cent stamps and some 20

cent stamps. He bought 37 stamps in all. How many of

each kind of stamp did he buy?

x + y = 37

.15x + .20y = 6.60 here is the problem

- .15x - .15y = -5.55 multiply eq 1 thru by -.15
_______________________
.05y = 1.05 subtract equations
___ _______
.05 .05 divide each side by .05

y = 21 divide and cancel

x + 21 = 37 replace y with 21

-21 -21 subtract 21 from each side
______________
x = 16 subtract

result: Sixteen 15 cent stamps. Twenty-one 20 cent stamps.

(9.) There were two models of calculators. One model sold

for $22.75 and the other model sold for 15.95. In

all, there were 31 calculators. The total sale of

these 31 calculators was $576.06 . How many of each

model were sold?

A + B = 31

22.75A + 15.95B = 576.05 here is the problem

-15.95A - 15.95B = -494.45 multiply eq 1 thru by -15.95
____________________________
6.80A = 81.60 subtract equations
_____ _____
6.80 6.80 divide each side by 6.80

A = 12 divide and cancel

12 + B = 31 replace A with 12

-12 -12 subtract 12 from each side
_________________
B = 19 subtract

result: 12 calculators and 19 calculators

(10.) A library charges a fixed amount for the first day that

a book is overdue and an additional charge for each day

thereafter. Sam paid $0.75 for one book that was 7 days

overdue and $1.95 for a book that was 19 days overdue.

Find the fixed charge and the charge for each additional

day.

f + 6a = 0.75

f + 18a = 1.95 here is the problem

f + 6a = 0.75 put this here
______________________
12a = 1.20 subtract equations
____ ___
12 12 divide each side by 12

a = .10 divide and cancel

f + 6(.10) = 0.75 replace a with 10

f + .60 = 0.75 multiply

- .60 -.60 subtract .60 from each side
_____________________
f = .15 subtract

result: The fixed charge is $0.15 and the additional charge

is $0.10 per day.