4. The larger of two numbers is 5 more than twice the smaller. If the smaller is subtracted from the larger, the result is 12. Find the two numbers.
x = the larger number
y = the smaller number
x = 2y + 5
x y = 12
2y + 5 y = 12
y + 5 = 12
y = 7
x = 2(7) + 5
x = 14 + 5
x = 19
The numbers are 19 and 7.
8. If tickets for a show cost $2.00 for adults and $1.50 for children, how many of each kind of ticket were sold if a total of 300 tickets were sold for $525?
|
|
Cost / ticket |
# tickets |
Total Cost |
|
Adult |
2 |
x |
2x |
|
Children |
1.50 |
y |
1.50y |
|
Total |
|
300 |
525 |
x + y = 300
2x + 1.50y = 525
x = 300 y
2(300 y) +1.50y = 525
600 2y + 1.50y = 525
-0.5y = -75
y = 150
x = 300 150 = 150
There were 150 adult tickets and 150 childrens tickets sold.
12. A woman earns $1350 in interest from two accounts in 1 year. If she has three times as much invested at 7% as she does at 6%, how much does she have in each account?
|
|
Amount Invested |
Interest Rate |
Interest Earned |
|
Acct. 1 |
x |
0.07 |
0.07x |
|
Acct. 2 |
y |
0.06 |
0.06y |
|
Total |
|
|
1350 |
x = 3y
0.07x + 0.06y = 1350
0.07(3y) + 0.06y = 1350
0.21y + 0.06y = 1350
0.27y = 1350
y = 5000
x = 3(5000) = 15000
She invested $15000 at 7% and $5000 at 6%.
16. How many ounces of 30% hydrochloric acid solution and 80% hydrochloric acid solution must be mixed to get 10 ounces of 50% hydrochloric acid solution?
|
|
Amt of Solution |
% HCl |
Amt of Pure HCl |
|
Soln 1 |
x |
0.30 |
0.30x |
|
Soln 2 |
y |
0.80 |
0.80y |
|
Total |
10 |
0.50 |
0.50(10) |
x + y = 10
0.30x + 0.80y = 0.50(10)
x = 10 y
0.30(10 y) + 0.80y = 0.50(10)
3 0.30y + 0.80y = 5
0.50y = 2
y = 4
x = 10 4
x = 6
Mix 6 ounces of 30% solution with 4 ounces of 80% solution.
20. A boat on a river travels 20 miles downstream in only 2 hours. It takes the same boat 6 hours to travel 12 miles upstream. What are the speed of the boat and the speed of the current?
|
|
Rate |
Time |
Distance |
|
Downstream |
b + c |
2 |
20 |
|
Upstream |
b c |
6 |
12 |
2(b + c) = 20
6(b c) = 12
2b + 2c = 20
6b 6c = 12
3(2b + 2c = 20)
6b 6c = 12
6b + 6c = 60
6b 6c = 12
12b = 72
b = 6
2(6) + 2c = 20
12 + 2c = 20
2c = 8
c = 4
The speed of the boat is 6 mph and the speed of the current is 4 mph.