Solving Word Problems - SAT Math

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown moulding to use as accent pieces. He needs $24\textup{ inches}$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $24\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$24\ inches:x\ feet\rightarrow \frac{24\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{24\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=24\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{24}{12}$

Solve.

$x=2 \ feet$

The carpenter needs $2 \ feet$ of material.

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Question

How many $\textup{centimeters}$ are in $13\textup { meters?}$

Answer

To solve this problem we can make proportions.

We know that $1\ centimeter= .01\ meters$ and we can use as our unknown.

$\frac{1\ centimeter}{.01\ meter}=\frac{x\ centimeters}{13\ meters}$

Next, we want to cross multiply and divide to isolate the on one side.

$1\ centimeter\times 13\ meters=.01\ meter\times x\ centimeters$

$\frac{1\ centimeter\times 13\ meters}{.01\ meter}= x\ centimeters$

The will cancel and we are left with $1,300\ centimeters$

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Question

How many $\textup { centimeters}$ are in $5\textup { meters?}$

Answer

To solve this problem we can make proportions.

We know that $1\ centimeter= .01\ meters$ and we can use as our unknown.

$\frac{1\ centimeter}{.01\ meter}=\frac{x\ centimeters}{5\ meters}$

Next, we want to cross multiply and divide to isolate the on one side.

$1\ centimeter\times 5\ meters=.01\ meter\times x\ centimeters$

$\frac{1\ centimeter\times 5\ meters}{.01\ meter}= x\ centimeters$

The will cancel and we are left with $500\ centimeters$

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown molding to use as accent pieces. He needs $18\textup{ inches}$ of the molding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $18\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$18\ inches:x\ feet\rightarrow \frac{18\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{18\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=18\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{18}{12}$

Solve.

$x=1 \tfrac{1}{2} \ feet$

The carpenter needs $1 \tfrac{1}{2} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown moulding to use as accent pieces. He needs $9\textup{ inches}$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $9 \ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$9\ inches:x\ feet\rightarrow \frac{9\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{9\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=9\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{9}{12}$

Solve.

$x=\frac{9}{12} \ feet$

Reduce.

$x=\frac{3}{4} \ feet$

The carpenter needs $\frac {3}{4} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown moulding to use as accent pieces. He needs $38\textup{ inches}$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $38\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$38\ inches:x\ feet\rightarrow \frac{38\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{38\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=38\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{38}{12}$

Solve.

$x=3 \tfrac{2}{12} \ feet$

Reduce.

$x=3 \tfrac{1}{6} \ feet$

The carpenter needs $3 \tfrac{1}{6} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown moulding to use as accent pieces. He needs $64\textup{ inches}$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $64\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$64\ inches:x\ feet\rightarrow \frac{64\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{64\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=64\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{64}{12}$

Solve.

$x=5 \tfrac{4}{12} \ feet$

Reduce.

$x=5 \tfrac{1}{3} \ feet$

The carpenter needs $5 \tfrac{1}{3} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\textup{ feet}$ of crown moulding to use as accent pieces. He needs $72\textup{ inches}$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $72\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$72\ inches:x\ feet\rightarrow \frac{72\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{72\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=72\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{72}{12}$

Solve.

$x=6 \ feet$

The carpenter needs $6 \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $98\ inches$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $98\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$98\ inches:x\ feet\rightarrow \frac{98\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{98\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=98\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{98}{12}$

Solve.

$x=8 \tfrac{2}{12} \ feet$

Reduce.

$x=8 \tfrac{1}{6} \ feet$

The carpenter needs $8 \tfrac{1}{6} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $124\ inches$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $124\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$124\ inches:x\ feet\rightarrow \frac{124\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{124\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=124\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{124}{12}$

Solve.

$x=10 \tfrac{4}{12} \ feet$

Reduce.

$x=10 \tfrac{1}{3} \ feet$

The carpenter needs $10 \tfrac{1}{3} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $39\ inches$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $39\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$39\ inches:x\ feet\rightarrow \frac{39\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{39\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=39\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{39}{12}$

Solve.

$x=3 \tfrac{3}{12} \ feet$

Reduce.

$x=3 \tfrac{1}{4} \ feet$

The carpenter needs $3 \tfrac{1}{4} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $87\ inches$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $87\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$87\ inches:x\ feet\rightarrow \frac{87\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{87\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=87\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{87}{12}$

Solve.

$x=7 \tfrac{3}{12} \ feet$

Reduce.

$x=7 \tfrac{1}{4} \ feet$

The carpenter needs $7 \tfrac{1}{4} \ feet$ of material.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $144\ inches$ of the moulding for the house. How many additional feet of the material will he need to purchase to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $144\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$144\ inches:x\ feet\rightarrow \frac{144\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{144\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=144\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{144}{12}$

Solve.

$x=12 \ feet$

The carpenter needs $12 \ feet$ of material. Since he already has he will need to purchase more to finish the project.

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Question

A carpenter is making a model house and he buys $8\ feet$ of crown moulding to use as accent pieces. He needs $164\ inches$ of the moulding for the house. How many feet of the material does he need to finish the model?

Answer

We can solve this problem using ratios. There are $12\ inches$ in $1\ foot$ . We can write this relationship as the following ratio:

$12\ inches:1\ foot\rightarrow \frac{12\ inches}{1\ foot}$

We know that the carpenter needs $164\ inches$ of material to finish the house. We can write this as a ratio using the variable to substitute the amount of feet.

$164\ inches:x\ feet\rightarrow \frac{164\ inches}{x\ feet}$

Now, we can solve for by creating a proportion using our two ratios.

$\frac{12\ inches}{1\ foot}=\frac{164\ inches}{x\ feet}$

Cross multiply and solve for .

$12\ inches \times (x\ feet)=164\ inches \times (1\ foot)$

Simplify.

Divide both sides by .

$\frac{12x}{12}=\frac{164}{12}$

Solve.

$x=13 \tfrac{8}{12} \ feet$

Reduce.

$x=13 \tfrac{2}{3} \ feet$

The carpenter needs $13 \tfrac{2}{3} \ feet$ of material.

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Question

Erin is making thirty shirts for her upcoming family reunion. At the reunion she is selling each shirt for $18 apiece. If each shirt cost her $10 apiece to make, how much profit does she make if she only sells 25 shirts at the reunion?

Answer

This problem involves two seperate multiplication problems. Erin will make $450 at the reunion but supplies cost her $300 to make the shirts. So her profit is $150.

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Question

Write as an equation:

"Ten added to the product of a number and three is equal to twice the number."

Answer

Let represent the unknown quantity.

The first expression:

"The product of a number and three" is three times this number, or

"Ten added to the product" is

The second expression:

"Twice the number" is two times the number, or

.

The desired equation is therefore

.

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Question

Write as an equation:

Five-sevenths of the difference of a number and nine is equal to forty.

Answer

"The difference of a number and nine" is the result of a subtraction of the two, so we write this as

"Five-sevenths of" this difference is the product of $\frac{5}{7}$ and this, or

$\frac{5}{7} (y- 9)$

This is equal to forty, so write the equation as

$\frac{5}{7} (y- 9) = 40$

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Question

Write as an equation:

Twice the sum of a number and ten is equal to the difference of the number and one half.

Answer

Let represent the unknown number.

"The sum of a number and ten" is the expression . "Twice" this sum is two times this expression, or

.

"The difference of the number and one half" is a subtraction of the two, or

$x - \frac{1}{2}$

Set these equal, and the desired equation is

$2 (x+10) = x - \frac{1}{2}$

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Question

If a rectangle possesses a width of $8\textup{ inches}$ and has a perimeter of $36\textup{ inches}$ , then what is the length?

Answer

In order to solve this problem, we need to recall the formula for perimeter of a rectangle:

We can substitute in our known values and solve for our unknown variable (i.e. length):

We want to isolate the to one side of the equation. In order to do this, we will first subtract from both sides of the equation.

$\frac{\begin{array}[b]{r}36=2l+16\ -16\ \ \ \ \ \ -16\end{array}}{\\20=2l}$

Next, we can divide each side by

$\frac{\begin{array}[b]{r}\frac{20}{2}=\frac{2l}{2}\\end{array}}{10=l}$

The length of the rectangle is $10\textup{ inches}$

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Question

If a rectangle possesses a width of $12\textup{ inches}$ and has a perimeter of $36\textup{ inches}$ , then what is the length?

Answer

In order to solve this problem, we need to recall the formula for perimeter of a rectangle:

We can substitute in our known values and solve for our unknown variable (i.e. length):

We want to isolate the to one side of the equation. In order to do this, we will first subtract from both sides of the equation.

$\frac{\begin{array}[b]{r}36=2l+24\ -24\ \ \ \ \ \ -24\end{array}}{\\12=2l}$

Next, we can divide each side by

$\frac{\begin{array}[b]{r}\frac{12}{2}=\frac{2l}{2}\\end{array}}{6=l}$

The length of the rectangle is $6\textup{ inches}$

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