7.1 – Proportions

Key Terms

• Proportion – An equation stating that two ratios are equal.
• A proportion of the form $\frac{a}{b}=\frac{c}{d}$ is true if the cross products are equal: ad = bc.
• Ratio – A comparison of two numbers. The ratio of a to b can be written a:b or $\frac{a}{b}$.
• Rational Expression – An expression that can be written as a fraction in which the numerator and denominator are polynomials.

Review

Equivalent Fractions
• Equivalent fractions have equal ratios
• $\frac{2400}{160}$ and $\frac{360}{24}$ are equal ratios.
• You can reduce each of these fractions by their greatest common multiple.
• $\frac{2400}{160}\div \frac{160}{160}=\frac{15}{1}$ and $\frac{360}{24}\div \frac{24}{24}=\frac{15}{1}$
• When solving real-world problems with ratios and proportions, ask  yourself, “what information is missing, and what do I want to find out?”
• Make sure you set up the ratios with the same categories in the numerator and denominator.
Correct

• $\frac{Old\; Width}{Old\; Height}$ and $\frac{New\; Width}{New\; Height}$
Incorrect

• $\frac{Old\; Width}{Old\; Height}$ and $\frac{New\; Height}{New\; Width}$

Notes

Proportions & Ratios
• Cross product – to find the answer using cross multiplication.
• When you cross multiply two equal ratios, the products are equal.

Changing the Size of a Photo Without Distorting It
• Suppose you want to shrink or enlarge an image.   You would want the faces and objects in the image to be proportional, wouldn’t you?  You don’t want anyone to look squished or stretched.  Watch this video to learn more about maintaining the original aspect ratio when reducing an image.

Examples

 Ex 1. Val is making rice. She knows that for every cup of uncooked rice, she needs to add 1.75 cups of water. If she has 3.5 cups of uncooked rice, how much water does she need? Setup: $\frac{1\;Cup \;of \;Rice}{1.75\;Cups \;of \;Water}=\frac{3.5\;Cups \;of \;Rice}{x\;Cups \;of \;Water}$ Numbers Only: $\frac{1}{1.75}=\frac{3.5}{x}$ Cross Multiply: 1x = (3.5)(1.75) Answer: x = 6.125 Ex 2. Brenda is 1.6m tall, and her shadow at 2pm is 2.3m long. How tall is the tree next to her if its shadow is 4.1m long? Setup: $\frac{Tree\; Height}{Tree\; Shadow}=\frac{Brenda$ Numbers Only: $\frac{x}{1.6}=\frac{4.1}{2.3}$ Cross Multiply: 2.3x = (1.6)(4.1) Simplify: $x=\frac{6.56}{2.3}$ Answer: 2.85m Ex 3. Solve for x Setup: $\frac{x}{6}=\frac{6}{4}$ Cross Multiply: $4x=(6)(6)$ Simplify: $4x=36$ Divide both sides by 4: $\frac{4x}{4}=\frac{36}{4}$ Answer: x = 9 Ex 4. Solve for x $\frac{x}{29}=\frac{2}{5.8}$ Cross Multiply: $5.8x=(2)(29)$ Simplify: $5.8x=58$ Divide both sides by 5.8: $\frac{5.8x}{5.8}=\frac{58}{5.8}$ Answer: x = 10 Ex 5. Tina worked 20 hours last week at her job at Sports Authority. She earned $155 for those hours. If her hourly rate stays the same, how many hours would she have to work to earn$217 next week? Since the hourly pay rate is the same, the ratios would be proportional. Set up a cross-product. Setup: $\frac{20}{155}=\frac{h}{217}$ Cross-Multiply: $(20)(217)=(155)(h)$ Simplify: $4340=155h$ Divide both sides by 155 to find the hours: $\frac{4340}{155}=\frac{155h}{155}$ Answer: 28 hours Ex 6. Maci has a family photograph with a width of 4 in and a height of 6 in. She wants to enlarge the photograph so that its width is 10 in. What will the height of the enlarged photograph be? Setup: $\frac{4}{6}=\frac{10}{x}$ Cross-Multiply: $(4)(x)=(6)(10)$ Simplify: $4x=60$ Divide both sides by 4 to find the height of the new photograph: $\frac{4x}{4}=\frac{60}{4}$ Answer: 15 inches

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