Elementary Algebra and Geometry for Schools

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HOME : Mathematics - Geometry

Some Standard Types of Word Problems

Age problems. Coin problems. Investment problems.

In this web pages we will learn to solve several types of word problems that follow certain patterns. These types of problems are often classified as " age" problems, "investment" problems, "coin" problems, and "lever" problems.

An introduction to age problems. If you are now 14 years old, how old will you be 6 years from now? How old were you 3 years ago? These are simple questions that you can readily answer, but they involve the general principle that is used in most age problems. Thus, if you know the present age of a person you can find what his age will be 6 years from now by adding 6 years to his present age. You can find what his age was 3 years ago by subtracting 3 years from his present age. In other words, in 5 years, Bill will be 5 years older than he is now; 4 years ago. Bill was 4 years younger than he is now.

EXERCISES: Introduction to age problems

1. A man is now 24 years old; how old will he be in 8 years?

2. A man is now x years old; how would you represent his age in 6 years?

3. A man is now y years old; how would you represent his age 4 years ago?

4. A boy was 12 years old 3 years ago; how old is he now?

5. A boywas 3x years old 3 years ago; how would you represent his age at the present time?

6. A boy was 2y years old 4 years ago; how would you represent his age 7 years ago?

7. A girl will be 18 years old 5 years from now; what is her present age?

8. A girl will be 5x years old 4 years from now; how would you represent her present age?

9. A girl will be 4y years old 6 years from now; how would you represent her age 2 years ago?

10. A boy's age 5 years from now will be 40-x, how would you represent his present age?

11. Jean is 4 years older than her brother. If her brother's present age is represented by x, how would you represent Jean's present age? How would you represent Jean's age 7 years ago?

12. A father is four times as old as his son. If the son's present age is represented by x, how would you represent the father's age at the present time? How would you represent the father's age 5 years hence (5 years from now)?

13. Six years ago a mother was four times as old as her daughter. If x represents the daughter's age 6 years ago, how would you represent the mother's age 6 years ago? How would you represent the mother's present age?

14. In 5 years, a father will be four times as old as his daughter. If x represents the daughter's age in 5 years, how would you represent the father's age in 5 years? How would you represent the father's present age?

15. The sum of the ages of Jim and John is 27 years. If x represents Jim's present age, how would you represent John's present age? How would you represent John's age 4 years ago?

EXERCISES: Arranging work in age problems

Copy the following boxes onto a paper and fill in the blank spaces with the correct data.

 

ILLUSTRATIVE EXAMPLE: Age problem

A boy is five times as old as his sister. In 9 years he will be twice as old as his sister will be. Find their present ages.

Solution. In this problem we are concerned with the ages of the boy and his sister at the present time, and in 9 years; therefore, we make a box with these headings.

Since the boy is now five times number of years in the sister's present age by x, and the number of years in the boy's present age by 5x, and put these expressions in their proper places in the box.

To represent the ages of the boy and his sister 9 years from now, we add 9 years to each of their present ages, filling in the second column in the box.

From the statement of the problem, "at that time (9 years from now), the boy will be twice as old as his sister." That is:

Referring to the box we proceed to answer the questions. The boy's present age, represented by 5x is 5(3) or 15 years. The sister's age, repreented by x, is 3 years. Check. Is the boy five times as old as his sister? Yes, since 15 is 5 times 3. "In nine years will he be twice as old as his sister?" Yes, since in 9 years he will be 15 + 9 or 24 years old; and in 9 years the sister will be 3 +9 or 12 years old; 24 is twice 12.

EXERCISES: Age problems

1. A father is five times as old as his son. In 4 years the sum of their ages will be 56 years. Find their present ages.

2. A father is now three times as old as his daughter. In 13 years he will be twice as old as his daughter will be then. Find their present ages.

3. The sum of the ages of a mother and her daughter is 34 years. In 7 years the mother will be three times as old as her daughter will be then. Find their present ages.

4 Four years ago a father was three times as old as his son was then. The sum of their present ages is 52 years. Find their present ages.

6. Five years ago a boy was twice as old as his sister was then. Six years from now the sum of their ages will be 43 years. Find their present ages.

6. Six years ago the sum of the ages of a father and his son was 55 years. Seven years from now the father will be twice as old as his son will be then. Find their present ages.

7. A mother's age is one year more than five times her son's age. In 19 years the mother will be twice as old as her son. Find their present ages.

8. Four years ago Alice was twice as old as Ruth, and Sally was one year older than Alice. Six years from now the sum of their ages will be 51 years. Find their present ages.

9. In 4 years James's age will be 9 years less than twice the age that John will be then. Seven years ago James was three times as old as John was then. Find their present ages.

10, Three years ago Philip was four times as old as David was then. Four years from now Philip's age will be one year more than twice the age that David will be then. Find their present ages.