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