"Number" Word Problems
Purplemath
"Number" word problems are usually fairly contrived, but they're also fairly standard, so you should learn how to handle them. After all, the point of these problems isn't their relation to "real life", but rather your ability to extract the math from the English.
These exercises are a great way to stretch your mental muscles, use what you know already, apply your logic (and common sense), and hippity-hoppity your way to the solution.
What is an example of solving a "number" word problem?
- The sum of two consecutive integers is 15. Find the numbers.
They've given me many pieces of information here.
- I'm adding (that is, summing) two things
- those things are numbers (like −4.628 and )
- the numbers are integers (like −3 and 6)
- the second number is 1 more than the first
- the result of the addition will be 15
How do I know that the second number will be larger than the first by 1? Because the two integers are "consecutive", which means "one right after the other, not skipping over anything between". (Examples of consecutive integers would be −12 and −11, 1 and 2, and 99 and 100.)
The "integers" are the number zero, the whole numbers, and the negatives of the whole numbers. In going from one integer to the next consecutive integer, I'll have gone up by one unit.
I need to figure out what are the two numbers that I'm adding. The second number is defined in terms of the first number, so I'll pick a variable to stand for this number that I don't yet know:
1st number: n
The second number is one more than the first, so my expression for the second number is:
2nd number: n + 1
I know that I'm supposed to add these two numbers, and that the result will be (in other words, I should set it equal to be) 15. This, along with my translation skills, allows me to create an equation, being the algebraic equivalent to "(this number) added to (the next number) is (fifteen)":
n + (n + 1) = 15
This is a linear equation that I can solve:
n + (n + 1) = 15
2n + 1 = 15
2n = 14
n = 7
The exercise did not ask me for the value of the variable n; it asked for the identity of two numbers. So my answer is not "n = 7"; the actual answer, taking into account the second number, too, is:
The numbers are 7 and 8.
It usually isn't required that you write your answer out like this; sometimes a very minimal "7, 8" is regarded as acceptible form. But the exercise asked me, in complete sentences, a question about two numbers; I feel like it's good form to answer that question in the form of a complete sentence.
What is an example of finding consecutive even/odd numbers?
- The product of two consecutive negative even integers is 24. Find the numbers.
I'll start with extracting the information they've given me.
- I'm multiplying (that is, finding the product of) two things
- those two things are numbers
- those two numbers are even
- those two numbers are integers
- those two numbers are negative
- one number is 2 more than the other
- when I multiply, I'll get 24
How do I know that one number will be 2 more than the other? Because these numbers are consecutive even integers; the "consecutive" part means "the one right after the other", and the "even" part means that the numbers are two units apart. (Examples of consecutive even integers are 10 and 12, −14 and −16, and 0 and 2.)
The second number is defined in terms of the first number, so I'll pick a variable for the first number. Then the second number will be two units more than this.
1st number: n
2nd number: n + 2
When I multiply these two numbers, I'm supposed to get 24. This gives me my equation:
(n)(n + 2) = 24
This is a quadratic equation that I can solve:
(n)(n + 2) = 24
n2 + 2n = 24
n2 + 2n − 24 = 0
(n + 6)(n − 4) = 0
This equation clearly has two solutions, being n = −6 and n = 4. Since the numbers I am looking for are negative, I can ignore the "4" solution value and instead use the n = −6 solution.
Then the next number, being larger than the first number by 2, must be n + 2 = −4, and my answer is:
The numbers are −6 and −4.
In the exercise above, each of the answers 4 and n = −6 was one of the solutions to the equation; the other solution to the equation had the opposite sign of the other answer to the exercise.
Do not assume that you can use both solutions if you just change the signs to be whatever you feel like. While this often "works", it does not always work, and it's sure to annoy your teacher. Instead, throw out invalid results, and solve properly for the valid ones.
- Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers?
The point of exercises like this is to give me practice in unwrapping and unwinding these words, and turning the words into algebraic expressions and equations. The point is in the solving, not in the relative "reality" of the problem. That said, how do I solve this? The best first step is to start labelling.
I need to find two numbers and, this time, they haven't given me any relationship between the two, like "two consecutive even integers". Since neither number is defined by the other, I'll need two letters to stand for the two unknowns. I'll need to remember to label the variables with their definitions.
the larger number: x
the smaller number: y
Now I can create expressions and then an equation for the first relationship they give me:
twice the larger: 2x
three more than five times the smaller: 5y + 3
relationship between ("is"): 2x = 5y + 3
And now for the other relationship they gave me:
four times the larger: 4x
three times the smaller: 3y
relationship between ("sum of"): 4x + 3y = 71
Now I have two equations in two variables:
2x = 5y + 3
4x + 3y = 71
I will solve, say, the first equation for x=:
Then I'll plug the right-hand side of this into the second equation in place of the "x":
4[ (5/2)y + (3/2) ] + 3y = 71
10y + 6 + 3y = 71
13y + 6 = 71
13y = 65
y = 65/13 = 5
Now that I have the value for y, I can back-solve for x:
x = (5/2)y + (3/2)
x = (5/2)(5) + (3/2)
x = (25/2) + (3/2)
x = 28/2 = 14
As always, I need to remember to answer the question that was actually asked. The solution here is not "x = 14", but is the following sentence:
The larger number is 14, and the smaller number is 5.
What are the steps for solving "number" word problems?
The steps for solving "number" word problems are these:
- Read the exercise through once; don't try to start solving it before you even know what it says.
- Figure out what you know (for instance, are you adding or multiplying?).
- Figure out what you don't know; this will probably be the values of numbers.
- Pick one or more useful variables for unknown(s) that you need to find.
- Use the variable(s) and the known information to create expressions.
- Use these expressions and the known information to create one or more equations.
- Solve the equation(s) for the unknown(s).
- Check your definition(s) for your variable(s).
- Use this/these definition(s) to state your answer in clear terms.
But more than any list, the trick to doing this type of problem is to label everything very explicitly. Until you become used to doing these, do not attempt to keep track of things in your head. Do as I did in this last example: clearly label every single step; make your meaning clear not only to the grader but to yourself. When you do this, these problems generally work out rather easily.
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