Linear Equation Word Problems
❶AGE PROBLEM
In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is68. How old is each one now?
This exercise refers not only to their present ages, but also to both their ages last year and their ages in three years, so labelling will be very important. I will label Miguel's present age as "m" and his grandfather's present age as "g". Then m + g = 68. Miguel's age "last year" was m – 1. His grandfather's age "in three more years" will be g + 3. The grandfather's "age three years from now" is six times Miguel's "age last year" or, in math: g + 3 = 6(m – 1)
This gives me two equations with two variables: m + g = 68 g + 3 = 6(m – 1)
Solving the first equation, I get m = 68 – g. (Note: It's okay to solve for "g = 68 – m", too. The problem will work out a bit differently in the middle, but the answer will be the same at the end.) I'll plug "68 – g" into the second equation in place of "m": g + 3 = 6m – 6 g + 3 = 6(68 – g) – 6 g + 3 = 408 – 6g – 6 g + 3 = 402 – 6g g + 6g = 402 – 3
7g = 399 g = 57
Since "g" stands for the grandfather's current age, then the grandfather is 57 years old. Sincem + g = 68, then m = 11, and Miguel is presently eleven years old.
SOURCE: http://www.purplemath.com/modules/ageprobs.htm
❷AGE PROBLEM
One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?
This problem refers to Heather's age two years in the future and three years in the past. So I'll pick a variable and label everything clearly: age now: H age two years from now: H + 2 age three years ago: H – 3
Now I need certain fractions of these ages: one-half of age two years from now: ( 1 2 )(H + 2) = H2 + 1 one-third of age three years ago: ( 13 )(H – 3) = H3 – 1
The sum of these two numbers is twenty, so I'll add them and set this equal to 20:
H2 + 1 + H3 – 1 =
In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is68. How old is each one now?
This exercise refers not only to their present ages, but also to both their ages last year and their ages in three years, so labelling will be very important. I will label Miguel's present age as "m" and his grandfather's present age as "g". Then m + g = 68. Miguel's age "last year" was m – 1. His grandfather's age "in three more years" will be g + 3. The grandfather's "age three years from now" is six times Miguel's "age last year" or, in math: g + 3 = 6(m – 1)
This gives me two equations with two variables: m + g = 68 g + 3 = 6(m – 1)
Solving the first equation, I get m = 68 – g. (Note: It's okay to solve for "g = 68 – m", too. The problem will work out a bit differently in the middle, but the answer will be the same at the end.) I'll plug "68 – g" into the second equation in place of "m": g + 3 = 6m – 6 g + 3 = 6(68 – g) – 6 g + 3 = 408 – 6g – 6 g + 3 = 402 – 6g g + 6g = 402 – 3
7g = 399 g = 57
Since "g" stands for the grandfather's current age, then the grandfather is 57 years old. Sincem + g = 68, then m = 11, and Miguel is presently eleven years old.
SOURCE: http://www.purplemath.com/modules/ageprobs.htm
❷AGE PROBLEM
One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?
This problem refers to Heather's age two years in the future and three years in the past. So I'll pick a variable and label everything clearly: age now: H age two years from now: H + 2 age three years ago: H – 3
Now I need certain fractions of these ages: one-half of age two years from now: ( 1 2 )(H + 2) = H2 + 1 one-third of age three years ago: ( 13 )(H – 3) = H3 – 1
The sum of these two numbers is twenty, so I'll add them and set this equal to 20:
H2 + 1 + H3 – 1 =