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Solve each of the following pairs of simultaneous equations, and check your answers in Exs. 1-5.
1 y 6 [Hint. Multiply the first by 3 and the second by 5.]
EXERCISES - APPLIED PROBLEMS
1. Find two numbers whose sum is 32 and whose difference is 6.
[HINT. Let x represent one of the numbers, and y the other. Then, from the statement of the problem, we must have x+y=32 and -y=6. Solve as in this Chapter, and check your answer.]
2. B had three times as much money as A. A then earned $5 and B spent $15, after which A had twice as much as B. How much did each have at first?
[Hint. Let x=the number of dollars A had, and y=the number of dollars B had. Then the two equations are 3 x=y and x+5= 2 (y-15). Explain.]
3. The sum of two numbers is 36, while their difference is 30. What are the numbers?
4. Find two numbers whose sum is – 55 and whose difference is – 5.
5. Find two numbers whose sum is 84 and such that the larger one is 48 more than three times the smaller one.
6. One half the sum of two numbers is 16, while one fourth their difference is – 2. What are the numbers ?
7. The sum of two numbers is zero, and their difference is 26. What are the numbers ?
8. The perimeter (distance around) of a certain rectangle is 2 inches more than three times the base, and the base is 14 times the height. What are the base and height?
9. An isosceles triangle is one which has two of its sides equal to each other. If one of the equal sides is 3 inches longer than the base,
Basele and the perimeter is 21 inches, what is the AL length of each of the three sides?
10. ABCD is a parallelogram whose sides AB, BC, CD, and DA, are all equal to each other. The perimeter is 25
inches more than the altitude, and four times the altitude is 2 inches less than twice the base. Find the base and altitude.
11. A father is twice as old as his son.
B Ten years ago he was three times as old Fig. 73.
as the son. Find the present age of each. [Hint. Let x=the number of years in the father's age, and y=the number of years in the son's age. Then, their ages 10 years ago were x-10 and y–10.]
12. In 10 years B will be as old as A, and in 20 years B will be s as old as A. Find the age of each.
13. A part of $2500 is invested at 6% and the remainder at 5%. The yearly income from both is $141. Find the amount in each investment.
14. A part of $5000 is invested at 7% and the remainder at 31%. If the 7% investment brings each year $193.75 more than the 31% investment, what is the amount of each?
15. A has a certain sum invested at 6% and B has another sum invested at the same rate. Their combined interest for one year is $300, but it takes B 8 years to receive as much as A receives in one year. How much has each invested ?
16. I have $1.20 and I want to go to the moving picture exhibition as many times as possible. If I pay my trolley fares and entrance fees and ride both ways each time, I can go 6 times; if I walk one way every time I can go 8 times. What is the trolley fare and what is the price of admission? 17. A and B together can do a piece of work in 12 days. After A had worked alone for 5 days, B finished the work in 26 days. In what time can each alone do the work?
[HINT. Let x=the number of days in which A can do it alone, and y=the number of days in which B can do it alone. Then the part A can do in one day is 1/x, and the part B can do in one day is 1/y. (Compare Ex. 21, p. 171.) So the equations become
1+1=1, and 5 + 25 = 1. Explain.
2 y Now solve as in § 131.]
18. A and B can do a certain piece of work in 16 days. They work together for four days, when B is left alone and completes the work in 3 days. In what time could each do it separately?
19. If 4 boys and 6 men can do a piece of work in 30 days, and 5 boys and 5 men can do the same work in 32 days, how long will it take 12 men to do the work?
20. Two weights just balance on a lever 13 feet long when the fulcrum is 8 feet from one end. If their positions be reversed, it is necessary to add 78 pounds to the lesser weight to restore the balance. What are the weights ? [Hint. See § 112.]
PROBLEMS ON SPECIAL TOPICS
I. NUMBERS AND DIGITS 21. If 1 is added to the numerator of a certain fraction, the value of the fraction becomes i; if 2 is added to the denominator, the value becomes 2. What is the fraction?
[Hint. Let x/y be the fraction.]
22. A certain fraction becomes equal to if 13 be added to both numerator and denominator, and it becomes equal to } if 27 be subtracted from both numerator and denominator. What is the fraction ?
23. A certain number of two digits equals four times the sum of the digits. What is the number? The digit in units' place is 3 greater than the digit in tens' place.
[Hint. Let x=the digit in tens place, and y=the digit in units' place. Then the number itself is 10 x+y. (Why?)
24. The sum of the digits of a certain number of two figures is 5, and if three times the units' digit is added to the number, the order of the digits is reversed. What is the number?
II. BILLS AND Coins 25. An errand boy went to the bank to deposit some bills, some of them being $1 bills and the rest $2 bills. If there were 38 bills in all and their combined value was $50, how many of each kind were there?
26. I have 15 coins, all silver dollars and quarters, whose value is $9.00. How many of each denomination are there?
27. The receipts from the sale of 300 tickets for a musical recital were $125. Adults paid 50 cents each, and children 25 cents each. How many tickets of each kind were sold ?
III. MIXTURES 28. A grocer wishes to make 50 pounds of coffee worth 32 cents per pound by mixing two other grades, one worth 26 cents per pound and the other 35 cents per pound. How much of each must he use?
[Hint. Let x= the amount to be used of the 26-cent grade, and y=the amount to be used of the 35-cent grade. Then x+y=50 and 26 x+35 y=50X32. Why?).
29. A grain dealer wishes to sell feed consisting of a mixture of corn and oats for $1.25 per hundred. Corn is worth $1.30 per hundred and oats $1.00. How many pounds of each must he put in each hundred of the mixture?