Solve.EXAMPLE 1Finding an Unknown NumberTwice a number, added to seven, is the same as thr...

Solve.

EXAMPLE 1

Finding an Unknown Number

Twice a number, added to seven, is the same as three subtracted from the number. Find the number.

Solution

Translate the sentence into an equation and solve.

To solve, begin by subtracting x from both sides to get all variable terms on one side.

Check the solution in the problem as it was originally stated. To do so, replace “number” in the sentence with −10. Twice “ −10 ” added to 7 is the same as 3 subtracted from “−10. ”

The unknown number is −10.

EXAMPLE 2

Finding an Unknown Number

Twice the sum of a number and 4 is the same as four times the number, decreased by 12. Find the number.

Solution

1. UNDERSTAND. Read and reread the problem. If we let

x = the unknown number, then

“the sum of a number and 4” translates to “x + 4” and “four times the number” translates to “4x.”

2. TRANSLATE.

3. SOLVE.

4. INTERPRET.

Check:  Check this solution in the problem as it was originally stated. To do so, replace “number” with 10. Twice the sum of “10” and 4 is 28, which is the same as 4 times “10” decreased by 12.

State:  The number is 10.

EXAMPLE 3

Finding the Length of a Board

Balsa wood sticks are commonly used for building models (for example, bridge models). A 48-inch balsa wood stick is to be cut into two pieces so that the longer piece is 3 times the shorter. Find the length of each piece.

Solution

1. UNDERSTAND the problem. To do so, read and reread the problem. You may also want to propose a solution. For example, if 10 inches represents the length of the shorter piece, then 31102 = 30 inches is the length of the longer piece, since it is 3 times the length of the shorter piece. This guess gives a total stick length of 10 inches + 30 inches = 40 inches, too short. However, the purpose of proposing a solution is not to guess correctly but to help understand the problem better and how to model it.

Since the length of the longer piece is given in terms of the length of the shorter piece, let’s let

2. TRANSLATE the problem. First, we write the equation in words.

3. SOLVE.

4. INTERPRET.

Check:  Check the solution in the stated problem. If the shorter piece of stick is 12 inches, the longer piece is 3 · (12 inches) = 36 inches, and the sum of the two pieces is 12 inches + 36 inches = 48 inches.

State:  The shorter piece of balsa wood is 12 inches, and the longer piece of balsa wood is 36 inches.

EXAMPLE 4

Finding the Number of Republican and Democratic Representatives

The 112th Congress began on January 3, 2011, and had a total of 435 Democrat and Republican representatives. There were 49 fewer Democratic representatives than Republican. Find the number of representatives from each party. (Source: Wikipedia.com)

Solution

1. UNDERSTAND the problem. Read and reread the problem. Let’s suppose that there were 200 Republican representatives. Since there were 49 fewer Democrats than Republicans, there must have been 200 − 49 = 151 Democrats. The total number of Republicans and Democrats was then 200 + 151 = 351. This is incorrect since the total should be 435, but we now have a better understanding of the problem. In general, if we let

2. TRANSLATE the problem. First, we write the equation in words.

3. SOLVE.

4. INTERPRET.

Check:  If there were 242 Republican representatives, then there were 242 − 49 = 193

Democratic representatives. The total number of representatives was then 242 + 193 = 435. The results check.

State:  There were 242 Republican and 193 Democratic representatives at the beginning of the 112th Congress.

EXAMPLE 5

Finding Angle Measures

If the two walls of the Vietnam Veterans Memorial in Washington, D.C., were connected, an isosceles triangle would be formed. The measure of the third angle is 97.5° more than the measure of either of the other two equal angles. Find the measure of the third angle. (Source: National Park Service)

Solution

1. UNDERSTAND. Read and reread the problem. We then draw a diagram (recall that an isosceles triangle has two angles with the same measure) and let

2. TRANSLATE. Recall that the sum of the measures of the angles of a triangle equals 180.

3. SOLVE.

4. INTERPRET.

Check:  If x = 27.5, then the measure of the third angle is x + 97.5 = 125. The sum of the angles is then 27.5 + 27.5 + 125 = 180, the correct sum.

State:  The third angle measures 125°.*

EXAMPLE 6

Some states have a single area code for the entire state. Two such states have area codes that are consecutive odd integers. If the sum of these integers is 1208, find the two area codes. (Source: North American Numbering Plan Administration)

Solution

1. UNDERSTAND. Read and reread the problem. If we let

2. TRANSLATE.

3. SOLVE.

4. INTERPRET.

Check:  If x = 603, then the next odd integer x + 2 = 603 + 2 = 605. Notice their sum, 603 + 605 = 1208, as needed.

State:  The area codes are 603 and 605.

Note:  New Hampshire’s area code is 603 and South Dakota’s area code is 605.

Beetles have the greatest number of different species. There are twenty times the number of beetle species as grasshopper species, and the total number of species for both is 420,000. Find the number of species for each type of insect.