Do you remember this problem that Julie had about the rainforest? Take a look.
Julie is amazed by all of the things that she is learning about the rainforest. One of the most interesting things that she has learned is that two-thirds of all of the fresh water on the planet is found in the Amazon River. Well, it isn’t exactly found in the Amazon, but in its basin, rivers, streams and tributaries. Julie is working on a drawing to show this. She draws the earth in one corner of the page and the Amazon River in the other corner of the page. As she reads on in her book on the Rainforest, she learns a new detail about the water of the Amazon. One-fifth of the water found in the Amazon River is found in its basin. Julie draws this on the page. She has the fraction two-thirds written near the top of the Amazon River and one-fifth written near its basin. “I wonder how much this actually is?” Julie thinks to herself. “How much is one-fifth of two-thirds?”
When Julie worked on this dilemma, she multiplied the two fractions.
Well, what if the fractions could be changed? What if x represented the amount of fresh water on the planet that is found in the Amazon River, and y represented the part found in the rest of the Amazon river not including the basin?
Could you write an expression and multiply these two quantities?
This Concept is all about expressions and products of fractions. Pay attention and you will know just how to do this at the end.
Guidance
An expression is a numerical phrase that combines numbers and operations but no equal sign.
There are two kinds of expressions. Numerical expressions include numbers and operations only. Variable (or algebraic) expressions include numbers, operations, and variables.
Includes | Examples | |
---|---|---|
numerical | numbers | 3 + 4 |
expressions | operations | \begin{align*}\frac{3}{4} \times \frac{2}{3}\end{align*} |
15.6 - 8 | ||
\begin{align*}4 \left (\frac{3}{4} \right )\end{align*} | ||
algebraic | numbers | \begin{align*}3 + x\end{align*} |
expressions | operations | \begin{align*}\frac{3}{4} \cdot \frac{b}{3}\end{align*} |
variables | \begin{align*}15.6 - q\end{align*} | |
\begin{align*}c \left (\frac{3}{4}\right )\end{align*} |
In this Concept we are going to be evaluating numerical and algebraic expressions. Let’s start with a numerical expression.
How do we evaluate a numerical expression?
Since a numerical expression includes numbers and operations, we simply perform the operation required to evaluate.
Evaluate \begin{align*}\left ( \frac{1}{4} \right ) \left ( \frac{3}{4} \right )\end{align*}
Notice that there are two sets of parentheses here. Remember that two sets of parentheses mean multiplication when they are next to each other.
We evaluate by multiplying and then simplifying or by simplifying first then multiplying.
\begin{align*}\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}\end{align*}
Our answer is in simplest form, so our work is complete.
What about algebraic expressions?
As you learn about algebra and higher levels of math, you will be working with algebraic expressions.
An algebraic expression has numbers and operations, but also variables. Often there are given values for the variables.
Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{3}{4}\end{align*}, and \begin{align*}y = \frac{1}{3}\end{align*}
To evaluate this expression, we substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression. The expression has \begin{align*}x\end{align*} and \begin{align*}y\end{align*} next to each other. When two variables are next to each other the operation is multiplication. We are going to multiply these fractions to evaluate the expression.
\begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
Next we apply what we learned in earlier Concepts to simplify first, if we can. Here we can simplify the threes. They simplify with the GCF of 3. Each three becomes a one.
\begin{align*}\frac{1}{4} \times \frac{1}{1} = \frac{1}{4}\end{align*}
Our answer is \begin{align*}\frac{1}{4}\end{align*}.
Solve a few of these on your own. Be sure that your answer is in simplest form.
Example A
Evaluate \begin{align*}\left ( \frac{4}{7} \right ) \left ( \frac{21}{28} \right )\end{align*}
Solution: \begin{align*}\frac{3}{7}\end{align*}
Example B
Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{3}{5}\end{align*} and \begin{align*}y\end{align*} is \begin{align*}\frac{10}{11}\end{align*}
Solution: \begin{align*}\frac{6}{11}\end{align*}
Example C
Evaluate \begin{align*}\left ( \frac{5}{9} \right ) \left ( \frac{45}{60} \right )\end{align*}
Solution: \begin{align*}\frac{5}{12}\end{align*}
Now back to the situation with Julie and the rainforest.
Julie is amazed by all of the things that she is learning about the rainforest. One of the most interesting things that she has learned is that two-thirds of all of the fresh water on the planet is found in the Amazon River. Well, it isn’t exactly found in the Amazon, but in its basin, rivers, streams and tributaries.
Julie is working on a drawing to show this. She draws the earth in one corner of the page and the Amazon River in the other corner of the page. As she reads on in her book on the Rainforest, she learns a new detail about the water of the Amazon. One-fifth of the water found in the Amazon River is found in its basin.
Julie draws this on the page. She has the fraction two-thirds written near the top of the Amazon River and one-fifth written near its basin.
“I wonder how much this actually is?” Julie thinks to herself. “How much is one-fifth of two-thirds?”
When Julie worked on this dilemma, she multiplied the two fractions.
Well, what if the fractions could be changed? What if x represented the amount of fresh water on the planet that is found in the Amazon River, and y represented the part found in the rest of the Amazon river not including the basin?
Could you write an expression and multiply these two quantities?
To multiply these two quantities, we first have to identify x and y.
\begin{align*}x = \frac{2}{3}\end{align*}
\begin{align*}y = \frac{4}{5}\end{align*}
If you are wondering where four - fifths came from, look back at the dilemma. One - fifth of the water is in the basin, so four -fifths is not.
Now we can multiply.
\begin{align*}\frac{2}{3} \times \frac{4}{5}\end{align*}
The answer is \begin{align*}\frac{8}{15}\end{align*}.
This means that eight - fifteenths of the earth's water is not found in the basin of the Amazon River.
Vocabulary
- Product
- the answer to a multiplication problem.
- Numerical Expression
- an expression that has numbers and operations.
- Algebraic Expression
- an expression that has numbers, operations and variables.
Guided Practice
Here is one for you to try on your own.
Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}y\end{align*} is \begin{align*}\frac{8}{12}\end{align*}
Answer
To complete this problem, we are going to multiply the two fractions. Be sure to simplify.
\begin{align*}\left ( \frac{2}{3} \right ) \left ( \frac{8}{12} \right )\end{align*}
\begin{align*}\left ( \frac{1}{3} \right ) \left ( \frac{4}{3} \right )\end{align*}
The answer is \begin{align*}\frac{4}{9}\end{align*}.
Video Review
These videos contain skills necessary for success when evaluating expressions involving the products of fractions.
Khan Academy Multiplying Fractions
James Sousa Multiplying Fractions
Practice
Directions: Evaluate each expression.
1. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{2}{3}\end{align*} and \begin{align*}y = \frac{6}{10}\end{align*}
2. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{1}{3}\end{align*} and \begin{align*}y = \frac{4}{10}\end{align*}
3. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{12}{13}\end{align*} and \begin{align*}y = \frac{2}{6}\end{align*}
4. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{1}{3}\end{align*} and \begin{align*}y = \frac{4}{5}\end{align*}
5. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{7}{9}\end{align*} and \begin{align*}y = \frac{3}{21}\end{align*}
6. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{4}{5}\end{align*} and \begin{align*}y = \frac{16}{20}\end{align*}
7. Evaluate \begin{align*}\left ( \frac{4}{6} \right ) \left ( \frac{1}{2} \right )\end{align*}
8. Evaluate \begin{align*}\left ( \frac{1}{9} \right ) \left ( \frac{6}{18} \right )\end{align*}
9. Evaluate \begin{align*}\left ( \frac{4}{9} \right ) \left ( \frac{1}{4} \right )\end{align*}
10. Evaluate \begin{align*}\left ( \frac{4}{11} \right ) \left ( \frac{11}{12} \right )\end{align*}
11. Evaluate \begin{align*}\left ( \frac{9}{10} \right ) \left ( \frac{5}{6} \right )\end{align*}
12. Evaluate \begin{align*}\left ( \frac{8}{9} \right ) \left ( \frac{3}{6} \right )\end{align*}
13. Evaluate \begin{align*}\left ( \frac{18}{19} \right ) \left ( \frac{3}{6} \right )\end{align*}
14. Evaluate \begin{align*}\left ( \frac{4}{9} \right ) \left ( \frac{36}{40} \right )\end{align*}
15. Evaluate \begin{align*}\left ( \frac{12}{14} \right ) \left ( \frac{7}{8} \right )\end{align*}