How to solve one-step equations with fractions in pre-algebra - Math

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Question

Solve for .

$p-$\frac{2}{5}$=3$

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Answer

Perform the same operation on both sides of the equation.

$p-$\frac{2}{5}$=3$

It will be easier to write the right side of the equation as a fraction.

$3=\frac{15}{5}$$

$p-$\frac{2}{5}$=\frac{15}{5}$$

Now, we add two-fifths to both sides of the equation.

$p-$\frac{2}{5}$+\frac{2}{5}$=\frac{15}{5}$+\frac{2}{5}$$

$p=\frac{17}{5}$$

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Question

is % of what number?

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Answer

To find the number of which is %, use this equation with % expressed as a fraction coefficient:

$$\frac{15}{100}$x = 30$

To solve this equation, multiply both sides of the equation by the reciprocal of the fraction on the left side, then reduce the result to simplest terms.

$($\frac{15}{100}$x) \times $\frac{100}{15}$ = (30) \times $\frac{100}{15}$$

$\rightarrow x = $\frac{3000}{15}$ \rightarrow x = 200$

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Question

Solve the equation for .

$\small $\frac{x}{2}$=7$

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Answer

$\small $\frac{x}{2}$=7$

Multiply both sides of the equation by $\small 2$ .

$\small $\frac{x}{2}$(2)=7(2)$

$\small x=14$

We can check our answer by plugging it back into the equation.

$$\frac{14}{2}$=?$

$$\frac{14}{2}$=7$

We know that our answer works.

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Question

Solve for if $x+\frac{1}{4}$=\frac{3}{8}$$

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Answer

To solve for we must get all of the numbers on the other side of the equation of .

To do this in a problem where a number is being added to , we must subtract the number from both sides of the equation.

In this case the number is $\frac{1}{4}$ so we subtract $\frac{1}{4}$ from each side of the equation to make it look like this $x+\frac{1}{4}$-$\frac{1}{4}$=\frac{3}{8}$-$\frac{1}{4}$$

To subtract fractions we must first ensure that we have the same denominator which is the bottom part of the fraction.

To do this we must find the least common multiple of the denominators.

The least common multiple is the smallest number that multiples of both of the denominators multiply to.

In this case the LCM is

We then multiply the numerator and denominator of $\frac{1}{4}$ by to get the same denominator because anything divided by itself is one so the fractions maintain their same value as the numbers change into the format we need to determine the answer.

To multiply a fraction you multiply the top number of the first fraction by the top number of the second fraction and the bottom number of the first fraction by the bottom number of the second fraction so it would look like this $\frac{1}{4}$*$\frac{2}{2}$=\frac{2}{8}$

After doing this we then subtract the first numerator (top part of the fraction) from the second numerator and place the result over the new denominator $x=\frac{3}{8}$-$\frac{2}{8}$$

The final answer is $x=\frac{1}{8}$$

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Question

A student is 5'4" tall. What is their height in cm?

$\small 1\ in =2.54\ cm$

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Answer

First, we need to convert the student's height to inches. There are 12 inches in one foot and the student is 5 feet and 4 inches tall. We need to covert feet to inches, and add 4.

$(5\ ft$\frac{12\ in}{1\ ft}$)+4\ in$

$60\ in+ 4\ in=64\ in$

So, the student is 64 inches tall. Now we need to convert to centimeters.

$64\ in$\frac{2.54\ cm}{1\ in}$=162.6\ cm$

The key to this kind of analysis is to make sure the units cancel correctly, leaving you with the units that you need.

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Question

Solve for $\small x$ .

$$\frac{x}{3}$=14$

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Answer

$$\frac{x}{3}$=14$

Multiply both sides by 3 to isolate $\small x$ .

$(3)$\frac{x}{3}$=(14)(3)$

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Question

Solve for if

$$\frac{x}{6}$=18$

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Answer

To solve for we must get all of the numbers on the other side of the equation of .

To do this in a problem where is being divided by a number, we must multiply both sides of the equation by the number.

In this case the number is so we multiply each side of the equation by to make it look like this

$6\cdot \left($\frac{x}{6}\right)=(18)\cdot 6$

The on the left side cancel and then we multiply $x=6\cdot 18$

The answer is .

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Question

Solve for if, $\frac{2x}{5}$=\frac{2}{3}$

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Answer

To solve for we must get all of the numbers on the other side of the equation as .

To do this in a problem where is being divided by a number, we must multiply both sides of the equation by the number.

Since it is a fraction we must multiply each side by the reciprocal like this

$\left($\frac{5}{2}\right)$\frac{2x}{5}$=\frac{2}{3}$\left($\frac{5}{2}\right)$

The numbers on the left cancel and we have

$x=\frac{2}{3}$\left($\frac{5}{2}\right)$

To multiply a fraction you multiply the top number of the first fraction by the top number of the second fraction and the bottom number of the first fraction by the bottom number of the second fraction.

We do this and find the answer is

$x=\frac{10}{6}$$

Reduce the fraction to get

$x=\frac{5}{3}$$

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Question

$$\frac{1}{3}$x=7$

What is ?

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Answer

To get rid of a fraction, we multiply by the reciprocal, so we take $$\frac{1}{3}$x=7$ and multiply both sides by $\frac{3}{1}$ :

$$\frac{1}{3}$x=7$

$\frac{3}{1}$$\frac{1}{3}$x=7$\frac{3}{1}$

Since $$\frac{3}{1}$$\frac{1}{3}$=1$ , we can simplify that equation to $x=7$\frac{3}{1}$$ .

Therefore, .

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Question

Solve for if, $\frac{5x}{3}$=\frac{8}{15}$

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Answer

To solve for we must get all of the numbers on the other side of the equation as .

To do this in a problem where is being divided by a number, we must multiply both sides of the equation by the number.

Since it is a fraction we must multiply each side by the reciprocal like this

$\left($\frac{3}{5}\right)\cdot $\frac{5x}{3}$=\frac{8}{15}$\cdot \left($\frac{3}{5}\right)$

The numbers on the left cancel and we have

$x=\left($\frac{8}{15}\right)\cdot \left($\frac{3}{5}\right)$

To multiply a fraction you multiply the top number of the first fraction by the top number of the second fraction and the bottom number of the first fraction by the bottom number of the second fraction.

We do this and find the answer is

$x=\frac{24}{75}$$

Reduce the fraction to get

$x=\frac{8}{25}$$ .

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Question

Solve for the value of $\small n$ .

$$\frac{n}{7}$=2$

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Answer

$$\frac{n}{7}$=2$

We need to isolate the variable. Multiply both sides by $\small 7$ .

$(7)$\frac{n}{7}$=(7)(2)$

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Question

$$\frac{1}{2}$x=6$

What is ?

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Answer

To get rid of a fraction, we multiply by the reciprocal. So we take $$\frac{1}{2}$x=6$ and multiply both sides by $\frac{2}{1}$ :

$$\frac{1}{2}$x=6$

$\frac{2}{1}$$\frac{1}{2}$x=6$\frac{2}{1}$

$\frac{2}{1}$ and $\frac{1}{2}$ cancel each other out, so we are left with $x=6*$\frac{2}{1}$$ or .

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Question

$$\frac{3}{5}$x=9$

What is ?

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Answer

To get rid of a fraction, we multiply by the reciprocal. So we take $$\frac{3}{5}$x=9$ and multiply both sides by $$\frac{5}${}3}$ :

$$\frac{3}{5}$x=9$

$\frac{5}{3}$$\frac{3}{5}$x=9$\frac{5}{3}$

Notice that $$\frac{5}${}3}$ and $\frac{3}{5}$ cancel out, leaving us with $x=9$\frac{5}{3}$$ .

At this point, you can either plug $9$\frac{5}{3}$$ into your calculator, or you can solve this in pieces.

We can do some manipulation to get: $9$\frac{5}{3}$=\frac{95}{3}$=\frac{9}{3}$5$

$$\frac{9}{3}$=3$ , so we can plug that into $x=\frac{9}{3}$5$ .

$x=\frac{9}{3}$*5$

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Question

$x=12*$\frac{1}{3}$$

What is ?

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Answer

To solve this problem, multiply across on the right side:

$x=12$\frac{1}{3}$=\frac{12}{1}$$\frac{1}{3}$=\frac{12}{3}$=4$

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Question

$$\frac{1}{9}$x=4$

What is ?

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Answer

To get rid of a fraction, we multiply by the reciprocal. So we take $$\frac{1}{9}$x=4$ and multiply both sides by $\frac{9}{1}$ :

$$\frac{1}{9}$x=4$

$\frac{9}{1}$$\frac{1}{9}$x=4$\frac{9}{1}$

Since $$\frac{9}{1}$$\frac{1}{9}$=1$ , we can simplify that equation to $x=4$\frac{9}{1}$$ .

Therefore, .

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Question

$$\frac{x}{4}$=2$

What is

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Answer

To get rid of the $\frac{x}{4}$ , we multiply both sides by :

$$\frac{x}{4}$=2$

$4$\frac{x}{4}$=24$

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Question

$x=\frac{8}{2}$$

What is ?

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Answer

To solve this problem we need to reduce $x=\frac{8}{2}$$ . Both the top and the bottom of $\frac{8}{2}$ are divisible by , so we can reduce it to $\frac{4}{1}$ . Anything divided by is itself, so $x=\frac{8}{2}$$ is the same as .

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Question

$x=9*$\frac{2}{3}$$

What is ?

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Answer

To solve, multiply the right side:

$x=9*$\frac{2}{3}$$

$x=\frac{18}{3}$$

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Question

$x=5*$\frac{1}{5}$$

What is ?

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Answer

For this problem, multiply across on the right side:

$x=5$\frac{1}{5}$$

$x=\frac{5}{1}$$\frac{1}{5}$=\frac{5}{5}$=1$

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Question

$x=13*$\frac{1}{2}$$

What is ?

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Answer

To solve this problem, multiply across: $x=13$\frac{1}{2}$=\frac{13}{1}$$\frac{1}{2}$=\frac{13}{2}$$ .

is a prime number, so we cannot reduce further. From here, covert $\frac{13}{2}$ into a mixed fraction: $x=6\t$\frac{1}{2}$$

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