Mathematics
Grade 7
15 min
Do the ratios form a proportion?
Do the ratios form a proportion?
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1
Introduction & Learning Objectives
Learning Objectives
Define ratio and proportion.
Identify equivalent ratios.
Apply the cross products property to check for proportionality.
Simplify ratios to their simplest form to compare them.
Determine if two given ratios form a proportion using various methods.
Solve real-world problems involving checking for proportionality.
Ever wondered how chefs scale recipes up or down without ruining the taste? 🧑🍳 It's all about keeping ingredients in balance, or 'in proportion'!
In this lesson, you'll discover how to tell if two ratios are equivalent, meaning they form a proportion. This fundamental skill is essential for understanding relationships between quantities and solving many practical problems in daily life.
Real-World Applications
Scaling r...
2
Key Concepts & Vocabulary
TermDefinitionExample
RatioA comparison of two quantities by division. It shows how much of one quantity there is compared to another.If there are 3 red apples and 2 green apples, the ratio of red to green apples is 3:2, or $\frac{3}{2}$.
ProportionAn equation that states that two ratios are equivalent (equal).$\frac{1}{2} = \frac{2}{4}$ is a proportion because the ratio 1:2 is equivalent to the ratio 2:4.
Equivalent RatiosRatios that represent the same relationship or value, even if the numbers are different.The ratios 1:2, 2:4, and 5:10 are all equivalent because they all simplify to 1:2.
Cross ProductsIn a proportion written as two fractions, the cross products are the products of the diagonal terms (numerator of one ratio multiplied by the denominator of the other).In the proportion $...
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Core Formulas
Cross Products Property
If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$. Conversely, if $a \times d = b \times c$, then $\frac{a}{b} = \frac{c}{d}$.
This property allows you to check if two ratios form a proportion without simplifying them. Multiply the numerator of the first ratio by the denominator of the second, and the denominator of the first by the numerator of the second. If these two products are equal, the ratios form a proportion.
Simplification Method
To check if $\frac{a}{b}$ and $\frac{c}{d}$ form a proportion, simplify both ratios to their simplest form. If their simplest forms are identical, then the ratios form a proportion.
Divide both parts of each ratio by their greatest common factor (GCF) until they cannot be simplified further. If the res...
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Challenging
If the ratio a/b is proportional to c/d, which of the following statements is ALWAYS true?
A.a × c = b × d
B.a/c = b/d
C.a + d = b + c
D.a - b = c - d
Challenging
A company tracks its sales. In Quarter 1, they sold 450 widgets and spent $90 on advertising. In Quarter 2, they sold 600 widgets and spent $125 on advertising. Is the ratio of widgets sold to advertising dollars proportional between the two quarters?
A.Yes, because sales and advertising both increased.
B.No, because the simplified ratio for Q1 is 5:1, while the simplified ratio for Q2 is 24:5.
C.Yes, because the cross products are equal.
D.No, because 600 - 450 is not equal to 125 - 90.
Challenging
Two painters are mixing the same shade of green paint, which requires a ratio of 3 parts blue paint to 5 parts yellow paint. Painter A mixes 12 parts blue and 20 parts yellow. Painter B mixes 10 parts blue and 15 parts yellow. Do BOTH painters' mixtures form a proportion with the original recipe?
A.Only Painter A's mixture is proportional.
B.Only Painter B's mixture is proportional.
C.Neither painter's mixture is proportional.
D.Both painters' mixtures are proportional.
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