Mathematics
Grade 8
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 and identify ratios and proportions.
Apply the cross-products property to determine if two ratios form a proportion.
Simplify ratios to their simplest form to compare them for proportionality.
Solve problems involving checking for proportionality in various contexts.
Explain the meaning of equivalent ratios and proportions.
Differentiate between ratios that form a proportion and those that do not.
Ever wondered if two different-sized photos have the exact same 'look' or if two recipes will taste the same even with different ingredient amounts? 🤔 That's often about whether their ratios form a proportion!
In this lesson, you will learn how to determine if two ratios are equivalent, which means they form a proportion. Understanding pr...
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Key Concepts & Vocabulary
TermDefinitionExample
RatioA comparison of two quantities by division. It can be written as a:b, a/b, or 'a to b'.If there are 3 red apples and 2 green apples, the ratio of red to green apples is 3:2 or 3/2.
ProportionAn equation stating that two ratios are equivalent. It shows that two ratios represent the same relationship.The equation 1/2 = 2/4 is a proportion because both ratios are equivalent.
Equivalent RatiosRatios that represent the same relationship or value, even if the numbers themselves are different.1:2 and 5:10 are equivalent ratios because both simplify to 1:2.
Cross ProductsIn a proportion a/b = c/d, the cross products are the results of multiplying the numerator of one ratio by the denominator of the other (a*d and b*c).For the proportion 1/2 = 2/4, the cross pr...
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Core Formulas
Cross-Products Property of Proportions
If $\frac{a}{b} = \frac{c}{d}$ (where $b \neq 0$ and $d \neq 0$), then $ad = bc$.
This is the most common and reliable method to check if two ratios form a proportion. If the product of the means (b*c) equals the product of the extremes (a*d), then the ratios are proportional. If the cross products are not equal, the ratios do not form a proportion.
Simplifying Ratios to Check Proportionality
To determine if $\frac{a}{b}$ and $\frac{c}{d}$ form a proportion, simplify both ratios to their simplest form. If the simplest forms are identical, then they form a proportion.
This method involves dividing both parts of each ratio by their greatest common factor (GCF) until no common factors remain. If the resulting simplified ratios are the same...
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Challenging
Do the ratios (x+2)/5 and (3x+6)/15 form a proportion for all values of x where the denominators are non-zero?
A.Yes, because the cross products are equal: 15(x+2) = 5(3x+6).
B.No, it only works for specific values of x.
C.No, because you cannot have variables in a proportion.
D.Yes, but only if x is a positive integer.
Challenging
In a school, the ratio of teachers to students is 3:70. In a neighboring school, there are 6 teachers and 145 students. Do the two schools have a proportional ratio of teachers to students?
A.Yes, because the number of teachers doubled and the number of students roughly doubled.
B.No, because the cross products are 435 and 420.
C.Yes, because the difference in students is large.
D.No, because 70 is not a multiple of 3.
Challenging
If a/b = c/d is a proportion, which of the following statements is NOT guaranteed to be true?
A.a/c = b/d (alternation property)
B.d/b = c/a (inversion and alternation)
C.(a+b)/b = (c+d)/d (componendo property)
D.a/d = b/c
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