Math Word Problems: Solving Number Problems Part 4

September 24, 2012 GB High School Algebra, High School Mathematics

This is the fourth and last part of the Number Word Problem Solving Series, the first subtopic in the Word Problem Solving Series. In this post, I will post more word problems with solutions. We start with the tenth problem in the series.

PROBLEM 10

There are three consecutive numbers. The sum of the first two is 20 more than the third. What are the numbers?

Solution

Let $x$ be the first number, $x + 1$ be the second number, and $x + 2$ be third number. As the problem says, the sum of the first two, $x +(x + 1)$ is $20$ more than the third. This means that if we add $20$ to the third number, $(x + 2) + 20$ , it will be equal to the sum of the sum of the first two. We now set up the equation

$x + (x + 1) = (x + 2) + 20$ .

Simplifying, we have $2x + 1 = x + 22$ . This means that $x = 21$ . So, the three consecutive numbers are $21$ , $22$ , and $23$ .

Checking the Answer

The sum of the first two integers $21 + 22 = 43$ . This is $20$ more than $23$ , the third number. So, we are correct.

PROBLEM 11

In a three digit number, the tens digit is half the hundreds digit. The ones digit is $1$ less than the tens digit. If the sum of the digits is $15$ , what is the number.

Solution

Let $x$ be hundreds digit. The tens digit is half of it, so it’s $\frac{1}{2}x$ . Then, the ones digit is $1$ less than the tens digit or $\frac{1}{2}x - 1$ . Since the sum of the three digits is $15$ , we can set up the equation

$x + \frac{1}{2}x + (\frac{1}{2}x - 1) = 15$ .

Multiplying both sides by $2$ , we have

$2x + x + (x - 2) = 30$ .

That gives us $4x - 2 = 30$ , $4x = 32$ and $x = 8$ . Therefore, the tens digit is $\frac{1}{2}(8) = 4$ and the ones digits is $4-1 = 3$ . So, the number is $843$ .

Checking the Answer

The tens digit $4$ is half of the hundreds digit $8$ . The ones digit is $3$ , one less than the tens digit.

PROBLEM 12

The sum of two numbers is $61$ . Three times the smaller is $13$ less than the larger.

Solution

Let $x$ be the smaller number and $61-x$ be the larger number. Three times the smaller, $3x$ is $13$ less than the larger. This means that if we add $13$ to $3x$ , it will be equal to the larger number which is $61 - x$ . So, we set up the equation