Practice Makes Perfect Algebra - Carolyn Wheater [29]
2. If the hot water tap is opened fully, it will fill the sink in 6 min. The cold water tap, fully open, can fill the same sink in 4 min. How long will it take to fill the sink if both taps are fully open?
3. Two machines in the school office can print final exams. The larger machine can print all the necessary exams in 12 h, but the smaller one would need 36 h to do the same job. How long will it take to print all the necessary exams if both machines are used?
4. Carrie can weed her vegetable garden in 2 h, and her younger brother, Andy, can do the same job in 4 h. How long will it take them to weed the garden if they work together?
5. Mr. Santiago can paint the living room in h, but if Ana helps him, they can get the job done in 90 min. How long would it take Ana to paint the room by herself?
6. Elena can complete a quilt in 6 mon, but if she and her mother work on it together, they can complete it in 2 mon. How long would it take Elena’s mom to complete the quilt by herself?
7. Every year, Mr. Song has to produce an annual report that fills many, many pages. If he types it himself, it takes him a week. If he gives it to his administrative assistant to type, she can do it in 3 days. How long will it take to get the annual report typed if both of them work on it?
8. If the drain pipe in the pool is opened, the pool will drain in 8 h. If, in addition to opening the drain, a pump is set up to put water out, the pool will be emptied in 5 h. How long would it take the pump, working alone, to empty the pool?
9. Alicia takes 6 h longer than David to complete a project. Working together, they can complete the project in 7.2 h. How long would it take Alicia to complete the project alone?
10. It takes Ron twice as long to mow his lawn with a push mower as it takes him to mow it with a power mower. He starts working with the power mower but after 45 min, he runs out of gas and has to finish the job with the push mower. It takes him an additional 45 min to finish the job with the push mower. How long would it have taken Ron to mow the entire lawn with the power mower?
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Exponential growth and decay
An exponent is shorthand for expressing repeated multiplication. When an unknown number is multiplied repeatedly, you get a power of a variable, like x5, and those powers combine into polynomials. When a number is used as a factor an unknown or variable number of times, the resulting power has the variable in the exponent position, like 5x. Exponential functions are functions of the form y = abx, where a and b are constants and b is greater than 0. Notice that the variable appears in the exponent position.
Compound interest
When you invest a sum of money, you expect that you’ll get your money back plus an additional amount, called interest. If you borrow money, you expect to repay the loan plus interest. The amount originally invested or borrowed is the principal, and the amount of interest earned or paid is determined by finding a percentage of the principal.
The interest paid on a loan or an investment can be calculated either as simple or compound interest. Simple interest is a one-time calculation: the amount loaned or invested times the interest rate per year times the number of years, or I = Prt. Compound interest is calculated at regular intervals—annually, semiannually, quarterly, monthly, or daily—and the interest for that period is added to the principal, and then earns interest along with the principal. If you invest $1000 at 5% per year for 2 years, simple interest would be I = 1000 × 0.05 × 2 = $100, for a total of $1100. If you invest the same $1000 at 5% per year for 2 years compounded annually, interest would be calculated at the end of the first year: I = 1000 × 0.05 × 1 = $50, and that $50 would be added to the original $1000. The interest for the second year would be calculated on $1050: I = 1050 × 0.05 × 1 = $52.50. After 2 years you will have earned $102.50 for a total of $1102.50.
After the first year, you had your original $1000 plus 5%, or 105%,