SAT Translating Word Problems

To solve problems that involve translation from words into math on the SAT, you need to know how to:

• manipulate algebraic expressions
• work with fractions and decimals
• work with percents
• work with inequalities
• translate words into math

$$\begin{array}{l l l} &\text{Addition} &+&\text{plus, added to, more than, greater than,}\\ &&&\text{sum, total, increased by}\\ &\text{Subtraction}&-&\text{minus, subtracted from, less than, fewer}\\ &&&\text{difference, decreased by, reduced by}\\ &\text{Multiplication}&\times, \cdot&\text{times, of, product, multiplied by}\\ &\text{Division}&/,\div, \frac{a}{b}, \% &\text{divided by, quotient, per, the ratio of}\\ &\text{Equals}&=&\text{is, will be, is equal to, is the same as,}\\ &&&\text{the result of, yields}\\ &\text{Power}&a^{2}, a^{3}, a^{4}&\text{the square of, squared, the cube of, cubed}\\ &&&\text{raised to the fourth power}\\ &\text{Root}&\sqrt{a}, \sqrt[3]{n}&\text{square-rooted, cube-rooted, the third root}\\ &\text{Multiplied by 2}& \times 2&\text{twice, doubled, two times,}\\ &&&\text{twice as much as}\\ &\text{Divided by 2:}&\div 2, \frac{a}{2} &\text{half of, half as much as, halved}\\ &&&\text{one-half times}\\ &\text{Inequality}&>&\text{more than, greater than}\\ &&\geq&\text{at least}\\ &&<&\text{fewer, less than}\\ &&\leq&\text{at most}\\ &\text{Unknown quantity}&x&\text{what, how many, how much, a number}\\ \end{array}$$

Four is $\underline{\quad \quad }$ less than seven.

Which of the following fills in the blank?

(A) $\$ Three
(B) $\$ Four
(C) $\$ Seven
(D) $\$ Eleven
(E) $\$ Twenty eight

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Correct Answer: A

Solution:

We replace the line with the variable $x$.

Translating the words into math, we get :

$$\underbrace{\text{Four}}_{4} \,\underbrace{ \text{is} }_{=} \, \underbrace{ \text{ x less than 7 } } _{ 7 - x }$$

We can now create an equation:

$$\begin{array} { r c l l } 4 & =& 7 - x &\quad \text{translate the words into math} \\ x + 4 & =& 7 &\quad \text{add x to both sides} \\ x & =& 7 - 4 &\quad \text{subtract 4 from both sides} \\ x & =& 3 &\quad \text{solve} \\ \end{array}$$

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Incorrect Choices:

(B)
Tip: Just because a number appears in the question doesn’t mean it is the answer.

(C)
Tip: Just because a number appears in the question doesn’t mean it is the answer.

(D)
Tip: Read the entire question carefully.
You will get this answer if you translate the phrase as

$$\underbrace{\text{Four}}_{4} \,\underbrace{ \text{is} }_{=} \, \underbrace{ x }_{x} \,\underbrace{\text{less than} }_{-} \, \underbrace{ \text{seven} }_{7}$$

(E)
This answer is offered to confuse you. It is the product of 4 and 7.

How much greater than $m-5$ is $m+7$?

(A) $\ \ 2$
(B) $\ \ 5$
(C) $\ \ 7$
(D) $\ \ 12$
(E) $\ \ 35$

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Correct Answer: D

Solution 1:

Let $m+7$ be $x$ greater than $m-5$. First we translate the words into math:

$\underbrace{\text{How much}}_{x} \quad \underbrace{\text{greater than}}_{+} \quad m-5\quad \underbrace{\text{is}}_{=}\quad m+7$?

We can now create an equation:

$$\begin{array}{r c l l l} x+m-5&=&m+7 &\quad \text{translate the words into math} &(1)\\ x-5&=&7 &\quad \text{subtract}\ m\ \text{from both sides} &(2)\\ x&=&7+5 &\quad \text{add}\ 5\ \text{to both sides} &(3)\\ x&=&12 &\quad 7+5=12 &(4)\\ \end{array}$$

Solution 2:

Tip: Look for short-cuts.
$m+7$ is greater than $m-5$. To find by how much, we subtract $m-5$ from $m+7$:

$$(m+7)-(m-5)=m+7-m+5=12$$

Solution 3:

Tip: Look for short-cuts.
Tip: Replace variables with numbers.
Let $m=0$. Then the question becomes: How much greater than $0-5$ is $0+7$, or, how much greater than $-5$ is $7$? The answer is $7-(-5)=12$.

Solution 4:

We can use the number line to solve the problem. Select any point for $m$. The point $7$ tick marks to the right of $m$ will be $m+7$. The point $5$ tick marks to the left of $m$ will be $m-5$. The number of segments between $m-5$ and $m+7$ is $12$, and therefore, $m+7$ is $12$ greater than $m-5$.

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Incorrect Choices:

(A)
Tip: Be careful with signs!
You might get this wrong answer if in step $(3)$ of Solution 1 you add $-5$ to both sides, not $5$, as shown:

$$\begin{array}{l c l l l} -5+x=7 &\quad &(2)\\ x=7\fbox{-}5 &\quad \text{mistake: added}\ -5\ \text{to both sides} &(3)\\ \end{array}$$

Similarly, in Solution 2 and Solution 3, you could make an error when distributing the negative sign:

Solution 2: $(m+7)-(m-5)=m+7-m\fbox{-}5=2$
Solution 3: $7-(-5)=7\fbox{-}5=2$

(B)
Tip: Just because a number appears in the question doesn’t mean it is the answer.

(C)
Tip: Just because a number appears in the question doesn’t mean it is the answer.

(E)
This answer is offered to confuse you. It is just the product of $5$ and $7$, the two numbers that appear in the prompt.

The difference of $5n$ and $2m$ squared is equal to the square root of the sum of $n$ squared and $m$ cubed.

Which of the following equations is an expression for the statement above?

(A) $\ \ 5n-(2m)\cdot {2} = \sqrt{n\cdot 2 + m\cdot 3}$
(B) $\ \ 5n-(2m)^{2} = \sqrt{n+m}$
(C) $\ \ 5n-(2m)^{2} = \sqrt{n^{2}}+\sqrt{m^{3}}$
(D) $\ \ 5n-(2m)^{2} = \sqrt{n^{2}+m^{3}}$
(E) $\ \ (5n-2m)^{2} = \sqrt{n^{2}+m^{3}}$

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Correct Answer: D

Solution:

Translating the words into math, we get:

The difference of $5n$ and $\underbrace{2m\ \text{squared}}_{(2m)^{2}}$ $\underbrace{\text{is equal to}}_{=}$ the square root of the sum of $\underbrace{n\ \text{squared}}_{n^{2}}$ and $\underbrace{m\ \text{cubed}}_{m^{3}}$. $$\begin{array}{r c l} \\ &\Downarrow&\\ \\ \underbrace{\text{The difference of}\ 5n\ \text{and}\ (2m)^{2}}_{5n-(2m)^{2}} &=& \text{the square root of}\ \underbrace{\text{the sum of}\ n^{2}\ \text{and}\ m^{3}}_{n^{2}+m^{3}}\\ \\ &\Downarrow&\\ \\ 5n-(2m)^{2} &=& \underbrace{\text{the square root of}\ n^{2}+m^{3}}_{\sqrt{n^{2}+m^{3}}}\\ \\ &\Downarrow&\\ \\ 5n-(2m)^{2} &=& \sqrt{n^{2}+m^{3}} \end{array}$$

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Incorrect Choices:

(A)
Squared means raised to the power of $2$. Cubed means raised to the power of $3$. The mistake here is that $2m$ is multiplied by $2$ instead of raised to the power of $2$, $n$ is multiplied by $2$ instead of raised to the power of $2$, and $m$ is multiplied by $3$ instead of raised to the power of $3$.

(B)
The right side of the equation ignores the instructions. $n$ should be squared, and $m$ should be cubed.

(C)
The mistake here is that the two terms, $n^{2}$ and $m^{3}$, are square-rooted individually, and then the sum of the square roots is found. However, the problem instructs that we find the sum $n^{2}+m^{3}$ first and then square-root it.

(E)
Here, the difference of $5n-2m$ is squared, but according to the problem, we should subtract $2m$ squared from $5n$.

If you thought these examples difficult and you need to review the material, these links will help:

• SAT Fractions and Decimals
• Combining Like Terms
• Distributive Property
  $a(b+c)=ab+ac$
• Simple Equations
• SAT Number Line

• Be careful with signs!
• Look for short-cuts.
• SAT General Tips