1.5 – Problem Solving

Problem Solving

A major reason to study mathematics is to bring the skills discussed into the real world. The beginning of this transition is to discuss word problems and to devise a strategy for solving them.

Problem Solving Strategy
Understand the problem. Read through the problem completely through. Ask yourself if you know what the problem wants you to solve. If not, re-read the problem. If a picture is appropriate, draw one here. Translate the word problem into a mathematical equation or equations that represent the problem. Solve the mathematical equation. Check your solution. Check the result in the original equation and ask yourself if the solution makes sense. State your solution Make sure that your answer the question that was asked, often times solving the equation does not necessarily mean you have solved the problem that was asked. Interpret the solution to the equation as a solution to the problem.

Problem Solving Strategy

Understand the problem. Read through the problem completely through. Ask yourself if you know what the problem wants you to solve. If not, re-read the problem. If a picture is appropriate, draw one here.
Translate the word problem into a mathematical equation or equations that represent the problem.
Solve the mathematical equation.
Check your solution. Check the result in the original equation and ask yourself if the solution makes sense.
State your solution Make sure that your answer the question that was asked, often times solving the equation does not necessarily mean you have solved the problem that was asked. Interpret the solution to the equation as a solution to the problem.

Eighteen increased by the product of four and a number is the same as eight decreased by six times the same number. Find the number
First, we must read through the problem. The problem is asking us to find the unknown number. So we write a mathematical equation. Translating the sentence gives us: $\underbrace{\text{Eighteen increased by the product of four and a number }}_{18+4x}$ $\underbrace{\text{ is the same as}}_{=} \underbrace{\text{ eight decreased by six times the same number.}}_{8-6x}$ $18+4x=8-6x$ We then solve the problem, using the general strategy provided in the previous section: $10x=-10$ $x=-1$ Thus, the unknown number is $-1$

Eighteen increased by the product of four and a number is the same as eight decreased by six times the same number. Find the number

First, we must read through the problem. The problem is asking us to find the unknown number. So we write a mathematical equation. Translating the sentence gives us:

$\underbrace{\text{Eighteen increased by the product of four and a number }}_{18+4x}$
$\underbrace{\text{ is the same as}}_{=} \underbrace{\text{ eight decreased by six times the same number.}}_{8-6x}$

$18+4x=8-6x$
We then solve the problem, using the general strategy provided in the previous section:

$10x=-10$

$x=-1$
Thus, the unknown number is $-1$

Twice the sum of a number and three is the same as six times the number decreased by two. Find the unknown number
Again, we must understand the problem and then translate. Then we will solve. $\underbrace{\text{Twice the sum of a number and three }}_{2(x+3)} \underbrace{\text{is the same as }}_{=} \underbrace{\text{ six times the number decreased by two}}_{6x-2}$ $2(x+3)=6x-2$ $2x+6=6x-2$ $-4x=-8$ $x=2$ This is the unknown number, so we are done. Checking the solution is left to the student.

Twice the sum of a number and three is the same as six times the number decreased by two. Find the unknown number

Again, we must understand the problem and then translate. Then we will solve.

$\underbrace{\text{Twice the sum of a number and three }}_{2(x+3)} \underbrace{\text{is the same as }}_{=} \underbrace{\text{ six times the number decreased by two}}_{6x-2}$
$2(x+3)=6x-2$
$2x+6=6x-2$
$-4x=-8$
$x=2$
This is the unknown number, so we are done. Checking the solution is left to the student.

A carpenter has a 60 inch board that he needs to cut into two pieces so that the longer piece is four times longer than the shorter. How long should each piece be?
Here, the question is asking for two values, the sizes of both cuts. – A picture may be helpful to see how to set up our equation. The board is to be cut in to two pieces. These pieces are not equal, so we can indicate this. It is not important that our drawing is extremely accurate, we simply need to indicate that there are two pieces: And some information is given about the two pieces. We know that the second piece is four times as long as the first. This means that the first piece can be labeled as $x$ , while the second can be labeled as $4x$ : The problem is written mathematically noting that the sum of the two pieces is known to be 60: $x+4x=60$ Simplifying gives: $5x=60$ Multiplication principle: $\frac{5x}{\color{red}5}=\frac{60}{\color{red}5}$ $x=12$ The small piece ( $x$ ) is 12, while the larger piece, ( $4x$ ) is $4\cdot12=48$ .

A carpenter has a 60 inch board that he needs to cut into two pieces so that the longer piece is four times longer than the shorter. How long should each piece be?

Here, the question is asking for two values, the sizes of both cuts. – A picture may be helpful to see how to set up our equation.

Rendered by QuickLaTeX.com

The board is to be cut in to two pieces. These pieces are not equal, so we can indicate this. It is not important that our drawing is extremely accurate, we simply need to indicate that there are two pieces:

And some information is given about the two pieces. We know that the second piece is four times as long as the first. This means that the first piece can be labeled as $x$ , while the second can be labeled as $4x$ :

Rendered by QuickLaTeX.com

The problem is written mathematically noting that the sum of the two pieces is known to be 60:

$x+4x=60$
Simplifying gives:
$5x=60$
Multiplication principle:
$\frac{5x}{\color{red}5}=\frac{60}{\color{red}5}$
$x=12$
The small piece ( $x$ ) is 12, while the larger piece, ( $4x$ ) is $4\cdot12=48$ .

A class of 250 students were assigned a project. If 22 fewer students completed the project than those who didn’t, how many completed the project?
We are not given the number of students who completed the project, but we have a relation between those who did and those who didn’t. We can say that $x$ students did not complete the project, and because 22 fewer students completed the project, we can say that $x-22$ students completed the project. We also know that since each student either finished the project or did not, then the sum of the two expressions will include every student, thus, we have: $x + (x-22) = 250$ And we can solve this equation. $x+x-22=250$ $2x-22=250$ $2x=272$ $x=136$ This, of course, is not the solution to the question that was asked. $x$ represents the number of students who did not complete the project, but we were asked to find the number of students who completed the project. This is $x-22$ or $136-22 = 114$

A class of 250 students were assigned a project. If 22 fewer students completed the project than those who didn’t, how many completed the project?

We are not given the number of students who completed the project, but we have a relation between those who did and those who didn’t. We can say that $x$ students did not complete the project, and because 22 fewer students completed the project, we can say that $x-22$ students completed the project.

We also know that since each student either finished the project or did not, then the sum of the two expressions will include every student, thus, we have:

$x + (x-22) = 250$
And we can solve this equation.
$x+x-22=250$
$2x-22=250$
$2x=272$
$x=136$

This, of course, is not the solution to the question that was asked. $x$ represents the number of students who did not complete the project, but we were asked to find the number of students who completed the project. This is $x-22$ or $136-22 = 114$

The first angle of a triangle is five degrees less than twice the measure of the second. The third is 25 less than three times the second. Find the measure of all three angles.
Again, a drawing helps visualize the problem. We are dealing with a triangle, and any generic triangle will do: Information is given about the angles. The second angle can be designated as $x$ , while the first angle is five less than double the second. As a mathematical expression, this is $2x-5$ . The third angle is twenty-five less than triple the second. This is $3x-25$ . We can label the triangle with these values. To set up an equation, we must know that the sum of the angles in the triangle is always $180^\circ$ . Then we obtain the equation: $(2x-5)+x+(3x-25)=180$ Solving gives: $2x-5+x+3x-25=180$ $6x-30=180$ $6x=210$ $x=35$ We have solved our equation, and must now translate this to a solution. Our problem asked us to find all three angles, so we have: Angle 2: $x=35$ Angle 1: $2x-5=2(35)-5=70-5=65$ Angle 3: $3(35)-25=105-25=80$

The first angle of a triangle is five degrees less than twice the measure of the second. The third is 25 less than three times the second. Find the measure of all three angles.

Again, a drawing helps visualize the problem. We are dealing with a triangle, and any generic triangle will do:

Rendered by QuickLaTeX.com

Information is given about the angles. The second angle can be designated as $x$ , while the first angle is five less than double the second. As a mathematical expression, this is $2x-5$ . The third angle is twenty-five less than triple the second. This is $3x-25$ . We can label the triangle with these values.

To set up an equation, we must know that the sum of the angles in the triangle is always $180^\circ$ . Then we obtain the equation:

$(2x-5)+x+(3x-25)=180$
Solving gives:
$2x-5+x+3x-25=180$
$6x-30=180$
$6x=210$
$x=35$

We have solved our equation, and must now translate this to a solution. Our problem asked us to find all three angles, so we have:

Angle 2: $x=35$
Angle 1: $2x-5=2(35)-5=70-5=65$
Angle 3: $3(35)-25=105-25=80$

The sum of three consecutive integers is 66. Find the numbers
Consecutive integers are numbers that are one apart. we can express the first unknown number as $x$ , and the second and third numbers as $x+1$ and $x+2$ . Since we know the sum of these three numbers is 66, we have the following equation: $x+(x+1)+(x+2)=66$ $x+x+1+x+2=66$ $3x+3=66$ $3x=63$ $x=21$ So we have the smallest number is 21, so the next two are 22 and 23. We can check and see that the sum of these numbers is indeed 66.

The sum of three consecutive integers is 66. Find the numbers

Consecutive integers are numbers that are one apart. we can express the first unknown number as $x$ , and the second and third numbers as $x+1$ and $x+2$ . Since we know the sum of these three numbers is 66, we have the following equation:
$x+(x+1)+(x+2)=66$
$x+x+1+x+2=66$
$3x+3=66$
$3x=63$
$x=21$
So we have the smallest number is 21, so the next two are 22 and 23. We can check and see that the sum of these numbers is indeed 66.

Four consecutive even numbers sum to 60. Find the numbers.
Since the numbers are consecutive even numbers, they are two apart. We can call the first number $x$ , then the next three numbers wil be $x+2$ , $x+4$ and $x+6$ . The sum of these is 60, so we have: $x+(x+2)+(x+4)+(x+6)=60$ $x+x+2+x+4+x+6=60$ $4x+12=60$ $4x=48$ $x=12$ Then the first consecutive even integer is 12. That makes the four numbers 12, 14, 16 and 18.

PreCalculus I

1.5 – Problem Solving

Problem Solving

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