Transpose and find the value of x.

51.

PROBLEMS.

1. Twice a certain number increased by 15 is equal to the number increased by 19. What is the number?

2. What number is that whose double exceeds the number by 12?

3. Ten times a certain equal to the number plus 5.

number diminished by 13 is What is the number?

4. What number diminished by 8 equals 6?

5. Six times a certain number plus 7 equals five times the number plus 12. What is the number?

6. Three boys together had 90 cents. The first had 10 cents more than the second, and the second had 1 cent How much had each ?

more than the third.

7. A father gave a certain sum to his youngest son, and 4 cents more to the next older, and 10 cents to the oldest. If he gave to all 20 cents, how much did he give to each ?

8. The greater of two numbers exceeds the less by 14, and the sum of the numbers is 34. What are the numbers?

9. A and B started in business, A furnishing $4000 more than B. Three times B's capital was then equal to

A's. How much did each furnish?

How much

10. A and B had the same sum of money. A gave B $4, and then B had double the amount A had left. had each at first?

A certain

11. A tourist rode 32 miles upon a bicycle. number of miles was down hill, twice as far plus 8 miles was level, and the distance up hill was as far as the distance on a level. How many miles did he travel upon

each kind of road.

12. A man has two horses, of unequal value, together worth $200. If he should put a saddle worth $30 on the poorer horse, the horse and saddle would together be equal in value to the better horse? What is the value of each?

13. Six hundred gallons of water are discharged into a cistern by 3 pipes. The second discharges 100 gallons more than the first, and the third discharges three times as much as the first. How many gallons are discharged by each pipe?

14. A drover being asked the number of his cattle said that if he had three times as many as he then had and 25 more, he would have 1000. How many cattle had he?

15. A man bought a watch and chain for $60. The watch cost 12 times as much as the chain lacking $5. What was the cost of each?

16. A tenement house contained 90 persons, men, women, and children. If there were 4 more men than women, and 10 more children than men and women together, how many were there of each ?

17. A steamer and its cargo are together worth $120,000. If the steamer lacks only $8400 of being worth twice as much as the cargo, what is the value of each?

18. A clerk's expenses are $400 per year, and his brother's are $600 per year. If the brother has three times as large an annual salary and he has left at the end of the year a sum equal to twice his brother's salary, what is the salary of each?

19. If a house and lot cost four times as much as the lot, and the house cost $2500 more than twice as much as the lot, what was the cost of each ?

MULTIPLICATION.

52. 1. What is the sum of 5 m +5 m + 5m? much is 3 times 5 m? 8 times 5 m?

Or, how

2. What is the sum of 8 xy + 8 xy + 8 xy + 8 xy? Or, how much is 4 times 8 xy? 10 times 8 xy?

3. How much is 6 times 7 be? Which quantity is the multiplier? Which is the multiplicand? What sign has the multiplier? What sign has the multiplicand? What

sign has the product?

4. When a positive quantity is multiplied by a positive quantity, what is the sign of the product?

5. If a vessel sails south 8 miles per hour, indicated by - 8 mi., how far will she sail in 5 hours?

the sign of the product?

What will be

6. How much is 4 times - 5 xy? 6 times - 6 ab? 7 times - 8 cd? What is the sign of the multiplier in each case? What is the sign of the multiplicand? What is the sign of the product? - 4 ab × 8 = ?

7. When a negative quantity is multiplied by a positive quantity, what is the sign of the product?

8. How does the product of 6 times 7 compare with the product of 7 times 6? What effect upon the product has it to change the order of the factors, when the numbers or quantities are abstract?

9. How, then, will the product of 3 a multiplied by +4 compare with the product of +4 multiplied by 3 a? What is the product? -5xy × 6= ? 6 × − 5 xy = ? - 7 ab × 4 = ? 4x-7 ab :

=

?

10. When a positive quantity is multiplied by a negative quantity, what is the sign of the product?

11. How much is 6 times - 3a? 2 times -3a? (6-2) times 3 a, or 6 times - 3 a, - 2 times 3 a?

12. Since 2 times 3a, or 6a, must be subtracted from 18 a to obtain the correct product, what will be the sign of 6 a after it is subtracted?

13. Since 2 times 3 a gives a product of +6 a, what may be inferred regarding the sign of the product when a negative quantity is multiplied by a negative quantity?

14. What is an exponent? What does it show? What does a mean? When a is multiplied by a2, how many times is a used as a factor to obtain the product? How many times, when a1 is multiplied by a5?

15. How, then, may the number of times a quantity is used as a factor in multiplication be determined from the exponents of the quantities in the expressions multiplied? How may the exponent of a quantity in the product be determined?

=

16. 3a2 x 6 ? 10 a5 x 5: = ? 20 a3 × 3 = ? 25 x2y × 2 =? How is the coefficient of the product determined from the coefficients of the factors, or from the multiplier and the multiplicand?

53. Multiplication is indicated in four ways: 1. By the sign ×, read multiplied by or times. Thus, a b shows that a is to be multiplied by b.