#algebra answers

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# WORD PROBLEMS

The solution here is not a number, because it will depend on the value of *b*. This is a type of “literal” equation, which is very common in algebra.

Example 3. The whole is equal to the sum of the parts.

The sum of two numbers is 84, and one of them is 12 more than the other. What are the two numbers?

**Solution.** In this problem, we are asked to find two numbers. Therefore, we must let *x* be one of them. Let *x*. then, be the first number.

We are told that the other number is 12 more, *x* + 12.

The problem states that their sum is 84:

The line over *x* + 12 is a grouping symbol called a *vinculum*. It saves us writing parentheses.

This is the first number. Therefore the other number is

*x* + 12 = 36 + 12 = 48.

The sum of 36 + 48 is 84.

Example 4. The sum of two consecutive numbers is 37. What are they?

**Solution**. Two consecutive numbers are like 8 and 9, or 51 and 52.

Let *x*. then, be the first number. Then the number after it is *x* + 1.

The problem states that their sum is 37:

The two numbers are 18 and 19.

Example 5. One number is 10 more than another. The sum of twice the smaller plus three times the larger, is 55. What are the two numbers?

**Solution.** Let *x* be the smaller number.

Then the larger number is 10 more: *x* + 10.

The problem states:

55 30 = 25.

That’s the smaller number. The larger number is 10 more: 15.

Example 6. Divide $80 among three people so that the second will have twice as much as the first, and the third will have $5 less than the second.

**Solution**. Again, we are asked to find more than one number. We must begin by letting *x* be how much the first person gets.

Then the second gets twice as much, 2*x* .

And the third gets $5 less than that, 2*x* 5.

Their sum is $80:

The sum of 17, 34, and 29 is in fact 80.

Example 7. Odd numbers. The sum of two consecutive odd numbers is 52. What are the two odd numbers?

**Solution**. First, an even number is a multiple of 2: 2, 4, 6, 8, and so on. It is conventional in algebra to represent an even number as 2*n*. where, by calling the variable ‘*n* ,’ it is understood that *n* will take whole number values: *n* = 1, 2, 3, 4, and so on.

An odd number is 1 more (or 1 less) than an even number. And so we represent an odd number as 2*n* + 1.

Let 2*n* + 1, then, be the first odd number. Then the next one will be 2 more — it will be 2*n* + 3. The problem states that their sum is 52:

Therefore the first odd number is 2** ** 12 + 1 = 25. And so the next one is 27. Their sum is 52.

Problem 1. Julie has $50, which is eight dollars more than twice what John has. How much has John? (Compare Example 1.)

First, what will you let *x* represent?

To see the answer, pass your mouse over the colored area.

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Do the problem yourself first!

The unknown number — which is how much that John has.

What is the equation?

Here is the solution:

Problem 2. Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore; how much did she spend there?

Here is the equation.

Here is the solution:

Problem 3. There are *b* black marbles. This is four more than twice the number of red marbles. How many red marbles are there? (Compare Example 2.)

Here is the equation.

Here is the solution:

Here is the solution:

Problem 10. Divide $79 among three people so that the second will have three times more than the first, and the third will have two dollars more than the second. (Compare Example 6.)

Here is the equation.

Here is the solution:

Problem 11. Divide $15.20 among three people so that the second will have one dollar more than the first, and the third will have $2.70 more than the second.

Here is the equation.

Here is the solution:

Problem 12. Two consecutive odd numbers are such that three times the first is 5 more than twice the second. What are those two odd numbers?

(See Example 7. where we represent an odd number as 2*n* + 1.)

**Solution**. Let the first odd number be 2*n* + 1.

Then the next one is 2*n* + 3 — because it will be 2 more.

The problem states, that is, the equation is: