Algebraic Addition in Problems. - Addition similar to that just explained is often required in solving problems. In our special equation drills (pages 30-37) the examples were easy either with or without equations. We are now going to take up problems which are easy if solved by equations, but hard when worked by arithmetic.

Example I. Helen is twice as old as John and the sum of their ages is 36 years. How old is each?

Hence John is 12 and Helen is 24 years old. Check in the problem.

Example II. Find three consecutive whole numbers whose sum is 36.

Solution. By the expression "consecutive whole numbers" is meant numbers that follow in order like 3, 4, and 5. Then if we represent the smallest number by n, the next is n + 1, and the third is n + 2.

The sum of n, n + 1, and n + 2, is 3 n + 3.

But the example says that the sum is also 36.

Therefore the required numbers are 11, 12, and 13.

Example III. The length of a rectangle is 5 feet more than its width, and its perimeter (distance around it) is 54 feet. What are its length and width?

Solution. Let x represent its width in feet.

Therefore the rectangle is 16 feet long and 11 feet wide.

Algebraic Short-Cuts Simplify Solutions

1. John is 3 times as old as Harry and the sum of their ages is 64. How old is each?

2. Mary had three times as much money as her brother. Together they had $24. How much had each?

3. The second angle of a triangle is 30° more than the smallest angle and the largest angle is 60° more than the

smallest. The sum of the angles of a triangle is 180°. How many degrees are there in each angle of this triangle?

4. Henry picked six more baskets of grapes than George. Together they picked 20 baskets. How many did each pick?

5. If one angle of a triangle is twice the second angle and the third angle is three times the second angle, how many degrees in each angle?

6. One acute angle of a right triangle (a triangle one of whose angles is equal to 90°) is 4 times the other acute angle. How many degrees in each of the acute angles?

7. A 42-inch cord is stretched around three pegs in the ground so as to form a triangle. Two sides are each three times as long as the shortest side. How long is each side of the triangle?

8. An isosceles triangle (a triangle two of whose sides are equal) has a perimeter of 40 inches. If the base is 4 inches

less than the sum of the two equal sides, what is the length of each side of the triangle?

9. One part of an 18-inch line is 3 times the other part. How many inches in each part?

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10. Find three consecutive odd whole numbers whose sum is 69. Hint: x = 1st number; x + 2 2d number. 11. Find four consecutive even whole numbers whose sum is 68.

12. Homer spent for his lunch 10 cents more than Jane, and Alfred spent as much as both Homer and Jane together. All the children together spent 80 cents. How much did each spend?

13. The longest side of a triangle is 3 inches more than the shortest side. The second side is 1 inch longer than the shortest side. The total length of the three sides is 25 inches. Find the length of each side.

14. A line 47 inches long is divided into three parts. The second part is 5 inches longer than the first part, and the third part is 12 inches longer than the first part. How long is each part?

15. Find the length and width of a rectangle which is 62 feet around, if the length is 7 feet more than the width.

16. A number is increased by of itself and the result is 462. What is the number? Hint: n + 3 n = 462.

17. I sell a vase for $6.25. If I make of the cost, what is the cost?

18. If 3.25 times a number plus 4.5 times the number equals 77.5, what is the number?

19. If 57 times a number minus of it equals 92, find the number.

20. If 7 times my money decreased by .75 of my money is $81, how much money have I?

21. After going of my trip, I have 60 miles yet to travel. How many miles had I to travel altogether?

22. Find four consecutive odd whole numbers whose sum is 56.

23. A steel rod 50 feet long must be cut into two pieces so that one piece is 12 feet longer than the other. Find the length of each piece.

24. Two trains start from the same station and run in opposite directions, one at the rate of 30 miles an hour, and the other at the rate of 35 miles an hour. In how many hours will they be 585 miles apart?

25. Five times a number is 10 more than three times the number. Find the number.

26. The second angle of a triangle is 45° more than the smallest angle. The third angle is three times the smallest angle. How many degrees in each angle if the sum of the angles of the triangle is 180°?

27. If 4 times a number increased by 4 equals the number increased by 16, find the number.