Problemas matemáticos

(Videos) Steps to Solving Word Problems

by mathman1024

This video by mathman1024 goes through the steps of solving word problems based on linear equations.

$\large{\text{Summary}}$

Let's be real for a second. Word problems are the worst. But sadly, you can't avoid them. Luckily there are tricks and steps you can learn to make solving them easier. It will still take lot's of practice but hopefully they can lead you in the right direction. The steps to solving word problems are:

1. READ and understand the problem. It may seem like common sense but it's very important to read the whole problem and understand what it's about.
2. DEFINE a variable. Figure out what the unknown in the problem is and assign a variable to it. For example it could be, $\text{Let }x=\text{the number}$ or $\text{Let }h=\text{the \# of hot dogs}$.
3. WRITE an equation. Check out how to translate words into expressions.
4. SOLVE the equation.
5. CHECK your answer.
6. STATE your answer. It's always good practice to state your answer at the end of a word problem. If the problem asks, "How many hot dogs did Lilly eat?" instead of saying $h=6$ write "Lilly ate $6$ hot dogs."

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He then goes over an example:

Five times a number plus nineteen is eighty-four. What is the number?

Step 1: READ the problem. It's okay if you need to read it multiple times to fully understand.

Step 2: DEFINE a variable. The problem is asking you to find a number. So we can say, $\text{Let }x=\text{the number}$.

Step 3: WRITE an equation. We know that we are multiplying a number, $x$, by five, so we can start with $5x$. We then add $19$ to that number, which gives us $5x+19$. This is all equal to $84$, so our full equation is $5x+19=84$.

Step 4: SOLVE the equation. $$\begin{align} 5x+19&=84\\ 5x+19\,\color{#8925AE}{-\,19}&=84\,\color{#8925AE}{-\,19}\\ 5x&=65\\ \frac{5x}{\color{#8925AE}{5}}&=\frac{65}{\color{#8925AE}{5}}\\ x&=13 \end{align}$$

Remember: whatever you do to one side of the equation, you have to do to the other. You may need to use the Addition and Subtraction Property of Equality or the Multiplication and Division Property of Equality.

Step 5: CHECK the answer. Check by plugging in $13$ everywhere we see $x$. $$\begin{align}5x+19&=84\\ 5({\color{#8925AE}{13}})+19&=84\\ 65+19&=84\\ 84&=84 \end{align}$$ This is a true statement, so our answer is correct.

Step 6: STATE your answer. $$\text{The number is }13.$$

(Video) Solving Word Problems

by mathman1024

This video works through a similar type of problem. We need to translate the problem

The sum of four times some number and forty-three is the same as three times the difference of the number and seventy-eight. Find the number.

We can use this statement to write an equation.

The left side of the equation is "the sum of four times some number and forty-three". We can make the unknown number $x$. Four times this number gives us $4x$. The sum of that value and $43$ is $4x+43$.

The right side of the equation is "three times the difference of the number and seventy-eight". This implies that our unknown number is the same number as the left side of the equation, so we'll use $x$ again. "Three times the distance of..." implies a use of distribution over parentheses with subtraction inside, so we can start with $3(x - \phantom{0})$. The last part is "difference of the number and seventy-eight", so we can subtract $78$ inside of the parentheses, $3(x-78)$.

Now we can set the two sides equal to each other and solve for $x$. $$4x+43 = 3(x-78)$$ First, we're going to distribute $3$ over $x-78$. $$4x+43 = 3x-234$$ Now, we can isolate $x$ on the left side by subtracting $3x$ and $43$ from both sides. $$\begin{align}4x+43 \,{\color{#8925AE}{- \, 3x}} &= 3x-234 \,{\color{#8925AE}{- \, 3x}}\\ x+43 \,{\color{#8925AE}{- \, 43}}&=234\,{\color{#8925AE}{- \, 43}}\\ x&=-277\end{align}$$

Our final answer is $x=-277$.

(Videos) Word Problems

by mathman1024

This video by mathman1024 works through a word problem based on the length of different boards. Problems like this get easier when you draw yourself a picture to work with.

$\large{\text{Summary}}$

The problem states:

A board that is 131 inches long is cut into two pieces. One piece is fourteen inches longer than twice the other. How long is each piece?

Sometimes it's helpful to draw a picture or diagram when working on word problems. Not every problem requires you to do so but in some, like this one, is can be beneficial. Let's draw our board before we set up the equation.

Now it says we're cutting it into two pieces. But notice that it doesn't say two equal pieces. This clues us in that we shouldn't cut the board down the middle. So let's just cut it randomly to the side.

We have two different pieces now but don't use two variables. Often in these problems you'll have one part you know nothing about and another part that's built off that. For example if you say, "Mira has $\text{\textdollar}10$ more than Sam," you don't know how much either has. But once you find out how much Sam has, you can find out how much Mira has.

In this problem, which piece do we know nothing about? The short piece. We know the long piece is fourteen inches longer than twice the other. Let's have our variable represent the length of the short piece. $$\text{Let }x=\text{length of short piece}$$

The other piece is fourteen inches longer than twice the other. So that means its length is $2x+14$.

This is a two-step equation word problem.

How do we relate these two pieces? Well the first sentence says a board that is $131$ inches long. So we know the combined length of these two pieces equals $131$ inches. In other words, $$\text{short piece}+\text{long piece}=\text{total board}$$ $$x+2x+14=131$$ From here we can solve like any other equation. $$\begin{align} x+2x+14&=131\\ 3x+14&=131\\ 3x+14\,\color{#8925AE}{-\,14}&=131\,\color{#8925AE}{-\,14}\\ 3x&=117\\ \frac{3x}{\color{#8925AE}{3}}&=\frac{117}{\color{#8925AE}{3}}\\ x&=39 \end{align}$$ Now $x$ is the length of the short piece. But the problem asks for the length of both pieces. So we need to plug our $x$-value into the expression for the long piece. $$2x+14\\ 2({\color{#8925AE}{39}})+14\\ 78+14\\ 92$$ So finally we state our answer. $$\text{The short piece is }39\text{ inches.}\\ \text{The long piece is }92\text{ inches.}$$

Here's another video with another word problem. This one is similar to the one above so try this yourself first.

The second angle of a triangle is four degrees less than the first angle, and the third angle is twice the first angle. What is the measure of each angle of the triangle?

We know that we have three angles in a triangle, so we can name them $m_1$, $m_2$, and $m_3$. We can set these equal to values.

• $m_1$ is going to be our base angle, so $m_1 = x$.
• $m_2$ is $4$ degrees less than $m_1$, so $m_2 = x-4$.
• $m_3$ is double $m_1$, so $m_3+2x$.

We also know that the angles of a triangle need to add to $180^{\circ}$, so $x+x-4+2x=180$.

We can solve this equation for $x$. First, we're going to combine like terms on the left side. $$\begin{align}x+x-4+2x&=180\\ 4x-4&=180 \end{align}$$

Now, we're going to add $4$ to both sides. $$\begin{align}4x-4&=180\\ 4x-4\,\color{#8925AE}{+\,4}&=180\,\color{#8925AE}{+\,4}\\ 4x&=184 \end{align}$$

Finally, we'll divide both sides by $4$ to get a value for $x$. $$\begin{align}4x&=184\\ \frac{4x}{\color{#8925AE}{4}}&=\frac{184}{\color{#8925AE}{4}}\\ x&=46 \end{align}$$

Now we can plug our $x$ value into the definitions for our angles. $$\begin{align}m_1&=x\\ m_1&=46^{\circ}\\ \\ m_2&=x-4\\ m_2&={\color{#8925AE}{46}}-4\\ m_2&=42^{\circ}\\ \\ m_3&=2x\\ m_3&=2({\color{#8925AE}{46}})\\ m_3&=92^{\circ} \end{align}$$

The last step is to test our angles to make sure they add up to $180$. $$46+42+92=180$$

So we have angles of $46^{\circ}, 42^{\circ},$ and $92^{\circ}$.

Word problems are math problems written in another language. In order to manage these problems, we need to take the words and translate them into mathematical expressions. Once we do that, we can solve these equations just like we normally would.

Let's take a look at an example of one of these two-step word problems:

On Saturdays, Jackson cuts grass and washes cars. He charges $\$ 15$ dollars per lawn and an additional fee for each car he washes. Last Saturday, Jackson mowed $5\text{ lawns}$ and washed $10\text{ cars}$. He made a total of $\$90$ How much does Jackson charge per car wash?

First, we need to figure out what the question is asking. What are we trying to find out?

We want to know how much Jackson charges per car wash. This means that the fee per car wash (the thing we don't know) will be represented as $x$.

We also know that the combined earnings for mowed lawns and washed cars is equal to $\$90$. This signals to us that we should put "$90$" on one side of the equals sign.

$$15(5) + 10x = 90$$

Notice that we've multiplied the fee for each service by the number of times Jackson completed it. So, $10x$ shows us that Jackson washed $10$ cars and earned $x$ dollars for each car. When we solve for $x$, we'll know how much Jackson charges per car wash.

To solve, all we need to do now is isolate $x$ on one side of the equation.

Remember, the problem asks us how much Jackson charges per car wash. So, while our value of $x = 1.5$ is a good start, we need to put it in a form that actually answers the question we were asked.

$x = 1.5$ means, Jackson charges $\$1.50$ per car wash

Let's check our work: $$15(5) + 10x = 90$$ Set $x=1.5$: $$15(5) + 10(1.5) = 90$$ $$75+15=90$$ $$90=90$$ We're correct!

(That's not a bad deal! Maybe Jackson can come wash my car...)

Let's try out another problem.

A high school marching band is traveling to a competition this weekend. The band has a total of $74\text{ members}$. The band will fill $3\text{ buses}$ and $5\text{ students}$ will follow in cars. How many students fit on each bus?

According to the word problem, we want to know the number of students that fit on each bus. This will be $x$ in our equation.

We also know that the total number of students in the band is $74$. Any time you see a "total," there's a good chance it'll be alone on one side of the equals sign.

$$3x + 5 = 74$$

Notice that the "$x$" represents the number of students on each of the three buses. To solve, all we need to do now is isolate $x$.

Remember, you need to go back to the original word problem to see what it is asking you, to determine if you need to change the form or units of your answer.

In this case, the problem asks how many students can fit on each bus. So, to make your answer totally correct, let's write it as: $23\text{ students}$.

Let's try another two-step word problem that has a slightly different format.

Maria bought a bag of $150\text{ jelly beans}$. She finds that there are twice as many green jelly beans as there are red, and twice as many blue as there are green. Maria also finds that there are half as many black jellybeans as there are red ones. How many red jellybeans are there in the bag?

We can start translating the word problem by recording what we know in terms of this variable $x$:

There are $x$ red jelly beans. So, $\text{red} = x$

There are $2x$ green jelly beans (twice as many as red). So, $\text{green} = 2x$

There are $4x$ blue jelly beans (twice as many as green). So, $\text{blue} = 4x$

There are $0.5x$ black jelly beans (half as many as red). So, $\text{black} = 0.5x$

Notice that since the problem asks us about how many red jelly beans there are, we had $x$ represent just red jelly beans, and set everything else equal to something that was relative to how many red jelly beans there are (in other words, relative to $x$).

Like before, we are given a total number (in this case, $150$ jelly beans). This signals to us that we should put $150$ alone on one side of the equals sign.

Try writing an equation that communicates the same information as the word problem, and then check below!

$$\text{red} + \text{green} + \text{blue} + \text{black} = 150\\\ x + 2x + 4x + 0.5x = 150$$

Now, simply combine all of the like terms (in this case, all of the terms on the left side of the equation) and solve for $x$!

$$\begin{align} x + 2x + 4x + 0.5x &= 150\\ 7.5x &= 150\\ \frac{7.5x}{\color{#8925AE}{7.5}} &= \frac{150}{\color{#8925AE}{7.5}}\\ x &= 20 \end{align}$$

Remember to check back with the original question to see exactly what you are being asked. In this case, we are asked how many red jelly beans are in the bag. To make our answer of $x=20$ even clearer, we can write: $20\text{ red jelly beans}$.

**Solving Word Problems **

To solve word problems, you need to translate the problem into numerical formulas. $\\$ Follow these steps to solve word problems.

1. Identify the unknown value in the problem and assign it a variable. $\\$ For example, $\text{ Let x= the number of dogs}$
2. Identify the known values (constants) in the equation.
3. Write an equation using the unknown and known variables.
4. Solve the equation. $\\$

You can check you answer by plugging it back into the unknown variable and solving. $\\$ Let’s look at an example.

Jack bought $35$ pencils at the beginning of the school semester. Throughout the semester, he lost some pencils. At the end of the semester, he had** $13$ pencils left**. How many pencils did Jack lose?

Step 1: Identify the unknown value in the problem and assign it a variable. The problem is asking you to find out how many pencils Jack lost. Assign this value $x$.

Step 2: Identify the known values (constants) in the equation. We know that Jack initially had $35$ pencils, and ended the semester with $13$ pencils.

Step 3: Write an equation using the unknown and known variables.
We know that we started with $35$ pencils, and since we lost $x$ pencils, we know that we subtract $x$ from $35$. This will equal to $13$, which is the final number of pencils that Jack is left with. So the equation would be: $$35-x = 13$$

Step 4: Solve the equation.

$$\begin{align} 35-x &= 13 \\ 35 -x \,\color{#8925AE}{-\,13} &= 13 \,\color{#8925AE}{-\,13} \\ 35 - 13 - x&=0 \\ 22 -x &=0 \\ 22 -x \,\color{#8925AE}{+\,x} &= 0 \,\color{#8925AE}{+\,x} \\ 22 &= x \\ x &= 22\\ \end{align}$$

Jack lost $22$ pencils during the semester.

We can check our answer by plugging $22$ back into the unknown variable $x$ and solve. $$\begin{align} 35-x &= 13 \\ 35 - ({\color{#8925AE}{22}}) &= 13 \\ 35 - 22 &= 13\\ 13 &= 13 \\ \end{align}$$

This statement is true, so our answer is correct.

Now, try an example by yourself.

Step 1: Identify the unknown value in the problem and assign it a variable. The problem is asking you to find out how many classes Leah is taking. Assign this variable $y$.

Step 2: Identify the known values (constants) in the equation. We know that each class gives about $30$ minutes, or $0.5$ hours of homework. We also know that she spent $2$ hours hanging out with her friends.

Step 3: Write an equation using the unknown and known variables.
We know that she takes $y$ classes and each class gives about $0.5$ hours of homework, so we can multiply $y$ and $0.5$ to represent the total amount of time she spent doing homework. She also spent $2$ hours hanging out with her friends, so we can add this to $y \cdot 0.5$. This will equal to $5.5$, which is the total amount hours she spent on schoolwork and hanging out with friends. So the equation would be: $$0.5 \cdot y + 2 = 5.5$$

Step 4: Solve the equation. $$\begin{align} 0.5 \cdot y + 2 &= 5.5\\ 0.5y + 2 &= 5.5 \\ 0.5 y + 2 \,\color{#8925AE}{-\,2} &= 5.5 \,\color{#8925AE}{-\,2} \\ 0.5y &= 5.5 -2 \\ 0.5y &= 3.5 \\ \dfrac{1}{2} y &= \dfrac{7}{2} \\ \dfrac{1}{2} y \,\color{#8925AE}{\cdot 2}&= \dfrac{7}{2}\,\color{#8925AE}{\cdot 2} \\ y &= 7 \\ \end{align}$$

Leah is taking $7$ classes. We can check our answer by plugging $7$ back into the unknown variable $y$ and solve. $$\begin{align} 0.5 \cdot y + 2 &= 5.5\\ 0.5 \cdot ({\color{#8925AE}{7}}) + 2 &= 5.5\\ 3.5+2 &= 5.5 \\ 5.5 &= 5.5 \end{align}$$

This statement is true, so our answer is correct.