algebra

ALGEBRAIC NOTATION.
1. Algebra is so much like arithmetic that all that you know about addition,
subtraction, multiplication, and division, the signs that you have been using
and the ways of working out problems, will be very useful to you in this study.
There are two things the introduction of which really makes all the di®erence
between arithmetic and algebra. One of these is the use of letters to represent
numbers, and you will see in the following exercises that this change makes the
solution of problems much easier.
Exercise I.
Illustrative Example. The sum of two numbers is 60, and the greater is four
times the less. What are the numbers?
Solution.
Let x= the less number;
then 4x= the greater number,
and 4x + x=60,
or 5x=60;
therefore x=12,
and 4x=48. The numbers are 12 and 48.
1. The greater of two numbers is twice the less, and the sum of the numbers
is 129. What are the numbers?
2. A man bought a horse and carriage for $500, paying three times as much
for the carriage as for the horse. How much did each cost?
3. Two brothers, counting their money, found that together they had $186,
and that John had ¯ve times as much as Charles. How much had each?
4. Divide the number 64 into two parts so that one part shall be seven times
the other.
5. A man walked 24 miles in a day. If he walked twice as far in the forenoon
as in the afternoon, how far did he walk in the afternoon?
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6. For 72 cents Martha bought some needles and thread, paying eight times
as much for the thread as for the needles. How much did she pay for each?
7. In a school there are 672 pupils. If there are twice as many boys as girls,
how many boys are there?
Illustrative Example. If the di®erence between two numbers is 48, and
one number is ¯ve times the other, what are the numbers?
Solution.
Let x= the less number;
then 5x= the greater number,
and 5x ¡ x=48,
or 4x=48;
therefore x=12,
and 5x=60.
The numbers are 12 and 60.
8. Find two numbers such that their di®erence is 250 and one is eleven times
the other.
9. James gathered 12 quarts of nuts more than Henry gathered. How many
did each gather if James gathered three times as many as Henry?
10. A house cost $2880 more than a lot of land, and ¯ve times the cost of the
lot equals the cost of the house. What was the cost of each?
11. Mr. A. is 48 years older than his son, but he is only three times as old.
How old is each?
12. Two farms di®er by 250 acres, and one is six times as large as the other.
How many acres in each?
13. William paid eight times as much for a dictionary as for a rhetoric. If the
di®erence in price was $6.30, how much did he pay for each?
14. The sum of two numbers is 4256, and one is 37 times as great as the other.
What are the numbers?
15. Aleck has 48 cents more than Arthur, and seven times Arthur's money
equals Aleck's. How much has each?
16. The sum of the ages of a mother and daughter is 32 years, and the age of
the mother is seven times that of the daughter. What is the age of each?
17. John's age is three times that of Mary, and he is 10 years older. What is
the age of each?
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