2.7: Mathematical Symbols to Represent Words
Rob is describing his weight training to his friend James. He said that when he started training he weighed 185 pounds. He gained 8 pounds in the first month of training. How much did he weigh?
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Khan Academy Problem Solving Word Problems 2
Guidance
Knowing how to translate key words from English into mathematical symbols is important in problem solving. The first step in any problem solving situation in mathematics is always to read the problem. Afterwards, translating the words into mathematical symbols is next.
Words such as gain, more, sum, total, increase, plus all mean to add. Words such as difference between, minus, decrease, less, fewer, and loss all mean to subtract. Words such as the product of, double \begin{align*}(2x)\end{align*}, twice \begin{align*}(2x)\end{align*}, triple \begin{align*}(3x)\end{align*}, a fraction of, a percent of, or times all mean to multiply. And finally, words such as the quotient of, divided equally, and per mean to divide.
Experience and practice with problem solving will help better acquaint you with the key words that translate into these operations.
Example A
What is the sum of five and seventeen?
Break apart the sentence. It is often helpful to underline the words before and after the word AND. Also, it is helpful to circle the mathematical symbol.
\begin{align*}& \text{What is the} \ \boxed{\text{sum}} \ \text{of} \ \underline{\text{five}} \ \text{and} \ \underline{\text{seventeen}}? \\ &\qquad \qquad \qquad \uparrow \qquad \quad \uparrow \qquad \quad \ \ \uparrow\\ & \qquad \qquad \qquad \ {\color{red}+} \qquad \ \ 5 \qquad \quad \ 17\end{align*}
Then translate the symbols together into a mathematical equation and solve it.
\begin{align*}5 {\color{red}+} 17 = 22\end{align*}
Example B
Thomas had twenty four dollars and after shopping his money decreased by four dollars.
Break apart the sentence. Underline the words before and after the word AND, and circle the mathematical symbol.
\begin{align*}& \text{Thomas had} \ \underline{\text{twenty-four}} \ \text{dollars and after shopping his money} \ \boxed{\text{decreased}} \ \text{by} \ \underline{\text{four}} \ \text{dollars.}\\ &\qquad \qquad \qquad \quad \uparrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \uparrow \qquad \qquad \ \ \uparrow\\ & \qquad \qquad \qquad \quad 24 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ {\color{red}-} \qquad \qquad \ 4\end{align*}
Then translate the symbols together into a mathematical equation and solve it.
\begin{align*}24 {\color{red}-}4 = 20\end{align*}
Therefore Thomas had $20.00 left after shopping.
Example C
Nick, Chris, and Jack are sharing a bag of jelly beans. There are 30 jelly beans for the three boys to share equally. How many would each get?
Again, break apart the sentence. Underline the words before and after the word AND, and circle the mathematical symbol.
\begin{align*}& \text{There are} \ \underline{30} \ \text{jelly beans for the} \ \underline{\text{three}} \ \text{boys to} \ \boxed{\text{share equally.}} \ \text{How many would each get?}\\ &\qquad \qquad \ \uparrow \qquad \qquad \qquad \qquad \qquad \uparrow \qquad \qquad \qquad \ \ \uparrow\\ &\qquad \qquad \ 30 \qquad \qquad \qquad \qquad \quad \ \ 3 \qquad \qquad \qquad \ \ {\color{red}\div}\end{align*}
\begin{align*}30 \div 3 = 10\end{align*}
Therefore each boy would get 10 jelly beans.
Concept Problem Revisited
Rob is describing his weight training to his friend James. He said that when he started training he weighed 185 pounds. He gained 8 pounds in the first month of training. How much did he weigh?
The word gain is the same as saying add.
Therefore Rob weighs \begin{align*}185 + 8 = 193 \ pounds\end{align*}.
Vocabulary
- Addition
- Words such as gain, more, sum, total, increase, plus all mean to addition or to add.
- Subtraction
- Words such as the difference between, minus, decrease, less, fewer, loss all mean subtraction or to subtract.
- Multiplication
- Words such as the product of, double \begin{align*}(2x)\end{align*}, twice \begin{align*}(2x)\end{align*}, triple \begin{align*}(3x)\end{align*}, fraction of, percent of, times all mean multiplication or to multiply.
- Division
Words such as the quotient of, divided equally, and per mean division or to divide.
Guided Practice
1. What is twelve increased by eighteen?
2. Joanne and Jillian were each going to share their babysitting money for the week. They made $45.00 in total. How much does each girl receive?
3. The number five is increased by seven. Three-fourths of this number is then decreased from twenty. What is the result?
Answers:
1. What is twelve increased by eighteen?
Break apart the sentence. Underline the words before and after the word AND, and circle the mathematical symbol.
\begin{align*}& \text{What is} \ \underline{\text{twelve}} \ \boxed{\text{increased}} \ \text{by} \ \underline{\text{eighteen}}?\\ &\qquad \qquad \ \uparrow \qquad \qquad \uparrow \qquad \qquad \quad \uparrow\\ &\qquad \qquad \ 12 \qquad \quad \ {\color{red}+} \qquad \qquad \quad 18\end{align*}
\begin{align*}12 + 18 = 30\end{align*}
2. Joanne and Jillian were each going to share their babysitting money for the week. They made $45.00 in total. How much does each girl receive?
Break apart the sentence. Underline the words before and after the word AND, and circle the mathematical symbol.
\begin{align*}& \underline{\text{Joanne and Jillian}} \ \text{were each going to} \ \boxed{\text{share}} \ \text{their babysitting money for the week. They made} \ \underline{\$45.00} \ \text{in total}?\\ &\qquad \qquad \uparrow \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ \uparrow \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \ \uparrow\\ &\qquad \qquad \ 2 \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ {\color{red}\div} \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \quad \ \ 45\end{align*}
\begin{align*}45 \div 2 = 22.5\end{align*}
Therefore each girl would get $22.50.
3. The number five is increased by seven. Three-fourths of this number is then decreased from twenty. What is the result?
Break apart the sentence. Underline the words before and after the word AND, and circle the mathematical symbol.
\begin{align*}& \text{The number} \ \underline{\text{five}} \ \text{is} \ \boxed{\text{increased}} \ \text{by} \ \underline{\text{seven}}. \ \boxed{\text{Three-fourths}} \ \text{of this} \ \underline{\text{number}} \ \text{is then} \ \boxed{\text{decreased}} \ \text{from} \ \underline{\text{twenty}}.\\ &\qquad \qquad \qquad \uparrow \qquad \quad \uparrow \qquad \qquad \quad \ \uparrow \qquad \qquad \quad \uparrow \qquad \qquad \qquad \quad \ \uparrow \qquad \qquad \qquad \quad \ \uparrow \qquad \qquad \qquad \uparrow\\ &\qquad \qquad \qquad 5 \qquad \quad {\color{red}+} \qquad \qquad \quad \ 7 \qquad \qquad \quad {\color{red}\times} \qquad \qquad \qquad \ \ 5+7 \qquad \qquad \qquad {\color{red}-} \qquad \qquad \quad \ 20\end{align*}
Step 1: \begin{align*}5 + 7 = 12\end{align*}
Step 2: \begin{align*}20- \frac{3}{4}(12) = 11\end{align*}
Practice
- Six less than fifty-three is what number?
- Twice the sum of eight and nine is what number?
- Twenty-five is diminished by four times five. What is the result?
- The product of five times four plus seven is what number?
- The sum of forty-four and fifty-two then divided by twelve results in what number?
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