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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?
If a man is now 24 years old, in 8 years he will be 24 + 8 = 32 years old.
2. A man is now x years old; how would you represent his age in 6 years?
If a man is now x years old, his age in 6 years would be x + 6 years old.
3. A man is now y years old; how would you represent his age 4 years ago?
If a man is now y years old, his age 4 years ago would be y - 4 years old.
4. A boy was 12 years old 3 years ago; how old is he now?
If a boy was 12 years old 3 years ago, it means that his current age is 12 + 3 = 15 years old. Therefore, the boy is now 15 years old.
5. A boy was 3x years old 3 years ago; how would you represent his age at the present time?
If a boy was 3x years old 3 years ago, his current age would be 3x + 3 years old.
6.
A boy was 2y years old 4 years ago; how would you represent his age 7 years ago?
If a boy was 2y years old 4 years ago, it means that his current age is 2y + 4 years old.
To represent his age 7 years ago, we can subtract 7 years from his current age:
Age 7 years ago = (2y + 4) - 7
Simplifying the expression, we get:
Age 7 years ago = 2y - 3 years old.
7. A girl will be 18 years old 5 years from now; what is her present age?
If a girl will be 18 years old 5 years from now, we can calculate her present age by subtracting 5 from 18:
Present age = 18 - 5 = 13 years old.
Therefore, the girl's present age is 13 years old.
8. A girl will be 5x years old 4 years from now; how would you represent her present age?
If a girl will be 5x years old 4 years from now, her current age would be 5x minus 4 years old.
9. A girl will be 4y years old 6 years from now; how would you represent her age 2 years ago?
If a girl will be 4y years old 6 years from now, it means that her current age is 4y - 6 years old.
To represent her age 2 years ago, we need to add 2 years to the age 6 years ago.
Age 6 years ago = (4y - 6) - 6 = 4y - 12
Age 2 years ago = (4y - 12) + 2
Simplifying the expression, we get:
Age 2 years ago = 4y - 10 years old.
10. A boy's age 5 years from now will be 40-x, how would you represent his present age?
If a boy's age 5 years from now will be 40-x, we can calculate his current age by subtracting 5 from 40-x:
Present age = (40-x) - 5 = 35 - x years old.
Therefore, the boy's present age is 35-x years old.
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?
If Jean's brother's present age is x, then Jean's present age can be represented by x + 4, since she is 4 years older than her brother.
To represent Jean's age 7 years ago, we need to subtract 7 from her current age:
Age 7 years ago = (x + 4) - 7
Simplifying the expression, we get:
Age 7 years ago = x - 3.
Therefore, Jean's present age is x + 4, and her age 7 years ago was x - 3.
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)?
If the father is four times as old as his son, then the father's present age can be represented by 4x, since he is four times as old as his son who is x years old.
To represent the father's age 5 years from now, we need to add 5 to his current age:
Father's age 5 years from now = 4x + 5
Therefore, the father's present age is 4x, and his age 5 years from now is 4x + 5.
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?
If x represents the daughter's age 6 years ago, then her present age is x + 6 years old.
Six years ago, the daughter's age was x, so her mother's age was 4x.
To represent the mother's age 6 years ago, we need to subtract 6 from her age 6 years ago:
Mother's age 6 years ago = 4x - 6
To represent the mother's present age, we need to add 6 to her age 6 years ago:
Mother's present age = (4x - 6) + 6
Simplifying the expression, we get:
Mother's present age = 4x
Therefore, the mother's age 6 years ago was 4x - 6, and her present age is 4x.
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?
If x represents the daughter's age in 5 years, then her father's age in 5 years can be represented by 4x, since he will be four times as old as his daughter.
To represent the father's present age, we need to subtract 5 from his age in 5 years:
Father's present age = 4x - 5.
Therefore, the father's age in 5 years is 4x, and his present age is 4x - 5.
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?
If x represents Jim's present age, then John's present age can be represented by (27 - x), since the sum of their ages is 27 years.
To represent John's age 4 years ago, we need to subtract 4 from his current age:
John's age 4 years ago = (27 - x) - 4
Simplifying the expression, we get:
John's age 4 years ago = 23 - x.
Therefore, John's present age is (27 - x), and his age 4 years ago was 23 - x.
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.
Let's represent the son's age by x. Then, the father's age can be represented by 5x, since he is five times as old as his son.
In 4 years, the son's age will be x + 4 and the father's age will be 5x + 4. The sum of their ages in 4 years will be (x + 4) + (5x + 4) = 6x + 8.
We know that in 4 years, the sum of their ages will be 56, so we can write an equation:
6x + 8 = 56
Simplifying and solving for x, we get:
6x = 48
x = 8
So the son's present age is 8 years old. Using the equation 5x, we can find the father's present age:
Father's present age = 5x = 5(8) = 40
Therefore, the son's present age is 8 years old, and the father's present age is 40 years old.
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.
Let's represent the daughter's present age by x. Then, the father's present age can be represented by 3x, since he is three times as old as his daughter.
In 13 years, the daughter's age will be x + 13 and the father's age will be 3x + 13. We know that in 13 years, the father will be twice as old as his daughter, so we can write an equation:
3x + 13 = 2(x + 13)
Simplifying and solving for x, we get:
x = 13
So the daughter's present age is 13 years old. Using the equation 3x, we can find the father's present age:
Father's present age = 3x = 3(13) = 39
Therefore, the daughter's present age is 13 years old, and the father's present age is 39 years old.
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.
Let's represent the daughter's present age by x. Then, the mother's present age can be represented by 34 - x, since the sum of their ages is 34.
In 7 years, the daughter's age will be x + 7 and the mother's age will be 34 - x + 7 = 41 - x. We know that in 7 years, the mother will be three times as old as her daughter, so we can write an equation:
41 - x = 3(x + 7)
Simplifying and solving for x, we get:
x = 10
So the daughter's present age is 10 years old. Using the equation 34 - x, we can find the mother's present age:
Mother's present age = 34 - x = 34 - 10 = 24
Therefore, the daughter's present age is 10 years old, and the mother's present age is 24 years old.
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.
Let's represent the son's present age by x. Then, the father's present age can be represented by the sum of his age four years ago (which was the son's age plus 4) and 4, since he is 4 years older now. So the father's present age can be represented by (x + 4) + 4 = x + 8.
We know that four years ago, the father was three times as old as his son was then, so we can write an equation:
Father's age four years ago = 3(Son's age four years ago)
(x + 4) - 4 = 3(x - 4)
Simplifying and solving for x, we get:
x = 12
So the son's present age is 12 years old. Using the equation x + 8, we can find the father's present age:
Father's present age = x + 8 = 12 + 8 = 20
Therefore, the son's present age is 12 years old, and the father's present age is 20 years old.
5. 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.
Let's use algebra to solve the problem.
Let x be the father's present age, and y be the son's present age.
From the first sentence, we have the equation:
x - 6 + y - 6 = 55
Simplifying and solving for x + y, we get:
x + y = 67
From the second sentence, we have the equation:
x + 7 = 2(y + 7)
Simplifying and substituting x + y = 67, we get:
67 - y + 7 = 2y + 14
Simplifying and solving for y, we get:
y = 22
Substituting y = 22 into x + y = 67, we get:
x = 45
Therefore, the father's present age is 45 and the son's present age is 22.
6. 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.
Let's use algebra to solve the problem.
Let x be the son's present age, and y be the mother's present age.
From the first sentence, we have the equation:
y = 5x + 1
From the second sentence, we have the equation:
y + 19 = 2(x + 19)
Simplifying and substituting y = 5x + 1, we get:
5x + 1 + 19 = 2x + 38
Simplifying and solving for x, we get:
x = 9
Substituting x = 9 into y = 5x + 1, we get:
y = 46
Therefore, the son's present age is 9 and the mother's present age is 46.
7. 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.
Let's use the following variables to represent their present ages:
- Alice's age: A
- Ruth's age: R
- Sally's age: S
From the problem, we can set up the following system of equations:
- Four years ago, Alice was twice as old as Ruth: A - 4 = 2(R - 4)
- Sally was one year older than Alice: S = A + 1
- Six years from now, the sum of their ages will be 51: (A + 6) + (R + 6) + (S + 6) = 51
We can simplify the first equation by expanding the right side:
- A - 4 = 2R - 8
- A = 2R - 4
Substituting this into the second equation gives us:
Now we can substitute both of these into the third equation and simplify:
- (2R - 4 + 6) + (R + 6) + (2R - 3 + 6) = 51
- 5R + 11 = 51
- 5R = 40
- R = 8
So Ruth is 8 years old now. Using the equation we derived earlier, we can find that Alice is:
- A = 2R - 4 = 2(8) - 4 = 12
And Sally is one year older than Alice, so:
Therefore, Alice is 12 years old, Ruth is 8 years old, and Sally is 13 years old.
8. 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.
Let's use algebra to solve the problem.
Let:
- Philip's current age = P
- David's current age = D
From the problem statement, we can set up two equations:
Equation 1: "Three years ago Philip was four times as old as David was then."
- (P - 3) = 4(D - 3) <-- Three years ago, Philip was P-3 and David was D-3
Equation 2: "Four years from now Philip's age will be one year more than twice the age that David will be then."
- (P + 4) = 2(D + 4) + 1 <-- Four years from now, Philip will be P+4 and David will be D+4
We now have two equations with two unknowns. We can solve for one variable and then substitute that into the other equation to solve for the other variable.
Simplifying Equation 1:
- P - 3 = 4D - 12
- P = 4D - 9
Substituting into Equation 2:
- (4D - 9 + 4) = 2(D + 4) + 1
- 4D - 5 = 2D + 9
- 2D = 14
- D = 7
Substituting D = 7 into Equation 1:
- P = 4D - 9
- P = 4(7) - 9
- P = 19
Therefore, Philip is currently 19 years old and David is currently 7 years old.
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