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# Mathematical Reasoning in Word Problems Help

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By McGraw-Hill Professional
Updated on Sep 26, 2011

## Mathematical Reasoning in Word Problems

### Consecutive Integers that Differ by One

Many problems require the student to use common sense to solve them—that is, mathematical reasoning. For instance, when a problem refers to consecutive integers, the student is expected to realize that any two consecutive integers differ by one. If two numbers are consecutive, normally x is set equal to the first and x + 1, the second.

#### Example 1:

The sum of two consecutive integers is 25. What are the numbers?

Let x = first number.

x + 1 = second number

Their sum is 25, so x + ( x + 1) = 25.

The first number is 12 and the second number is x + 1 = 12 + 1 = 13.

#### Example 2:

The sum of three consecutive integers is 27. What are the numbers?

Let x = first number.

x + 1 = second number

x + 2 = third number

Their sum is 27, so x + ( x + 1) + ( x + 2) = 27.

The first number is 8; the second is x + 1 = 8 + 1 = 9; the third is x + 2 = 8 + 2 = 10.

Find practice problems and solutions at Mathematical Reasoning in Word Problems Practice Problems - Set 1.

## Consecutive Integers that Differ by More than One

The sum of two numbers is 70. One number is eight more than the other. What are the two numbers?

Problems such as this are similar to the above in that we are looking for two or more numbers and we have a little information about how far apart the numbers are. In the problems above, the numbers differed by one. Here, two numbers differ by eight.

Let x = first number. (The term “first” is used because it is the first number we are looking for; it is not necessarily the “first” in order.) The other number is eight more than this, so x + 8 represents the other number. Their sum is 70, so x + ( x + 8) = 70.

The numbers are 31 and x + 8 = 39.

#### Example

The sum of two numbers is 63. One of the numbers is twice the other.

Let x = first number.

2 x = other number

Their sum is 63, so x + 2 x = 63.

The numbers are 21 and 2 x = 2(21) = 42.

Find practice problems and solutions at Mathematical Reasoning in Word Problems Practice Problems - Set 2.

## Consecutive Integers and Multiplying by Quantities

The difference between two numbers is 13. Twice the smaller plus three times the larger is 129.

If the difference between two numbers is 13, then one of the numbers is 13 more than the other. The statement “The difference between two numbers is 13,” could have been given as, “One number is 13 more than the other.” As before, let x represent the first number. Then, x + 13 represents the other. “Twice the smaller” means “2 x ” ( x is the smaller quantity because the other quantity is 13 more than x ). Three times the larger number is 3( x + 13). “Twice the smaller plus three times the larger is 129” becomes 2 x + 3( x + 13) = 129.

The numbers are 18 and x + 13 = 18 + 13 = 31.

#### Example

The sum of two numbers is 14. Three times the smaller plus twice the larger is 33. What are the two numbers?

Let x represent the smaller number. How can we represent the larger number? We know that the sum of the smaller number and larger number is 14. Let “?” represent the larger number and we’ll get “?” in terms of x .

So, 14 − x is the larger number. Three times the smaller is 3 x . Twice the larger is 2(14 – x ). Their sum is 33, so 3 x + 2(14 − x ) = 33.

 3 x + 2(14 – x ) = 33 3 x + 28 – 2 x = 33 x + 28 = 33 – 28 – 28 x = 5

The smaller number is 5 and the larger is 14 – x = 14 − 5 = 9.

Find practice problems and solutions at Mathematical Reasoning in Word Problems Practice Problems - Set 3.

More practice problems for this concept can be found at: Algebra Word Problems Practice Test.

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