Solving Proportions using Cross-Multiplication
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Interactive Audio Lesson
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Understanding Proportions
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Today, we're diving into proportions! Does anyone know what a proportion is?
Isn't it how two ratios compare?
Exactly! A proportion states that two ratios are equal, like \( \frac{a}{b} = \frac{c}{d} \). Now, what do you think cross-multiplication implies?
I think it's about multiplying the diagonals?
Yes! We multiply diagonally, setting up the equation \( a \times d = b \times c \). Remember: **Cross means multiply!**
Solving a Proportion
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Letβs solve one: Suppose \( \frac{3}{x} = \frac{6}{12} \). How do we start?
We can cross-multiply, right?
Correct! That gives us \( 3 \times 12 = 6 \times x \). What is that?
Thatβs \( 36 = 6x \)!
Great! Now, how do we isolate \( x \)?
Divide both sides by 6 to get \( x = 6 \)!
Exactly! Well done team!
Applying Cross-Multiplication in Real-Life Scenarios
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Now, can anyone think of a real-life application of proportions?
Maybe in cooking, like adjusting recipe portions?
Awesome example! If a recipe serves 4 with 2 cups of flour, and we want to serve 10, how do we set up a proportion to find out how much flour we need?
We can set it up as \( \frac{2}{4} = \frac{x}{10} \).
Exactly! Letβs cross-multiply. What do we get?
That gives us \( 2 \times 10 = 4x \), which simplifies to 20 = 4x!
Great job! Now, how do we find \( x \)?
Divide by 4: \( x = 5 \) cups of flour!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, students learn the concept of proportions and the cross-multiplication method as a means to solve them. The significance of equal ratios and how to apply cross-multiplication to find unknown values is thoroughly discussed.
Detailed
Solving Proportions using Cross-Multiplication
Proportions are equations that express the equality of two ratios. They can be represented as
\[ a/b = c/d \]
where \(a\), \(b\), \(c\), and \(d\) are numbers, with \(b \neq 0\) and \(d \neq 0\). To find an unknown value in a proportion, we can utilize cross-multiplication, a straightforward algebraic technique.
With cross-multiplication, we multiply the numerator of one ratio by the denominator of the other ratio. This results in the equation:
\[ a \times d = b \times c \]
This technique not only simplifies the calculation but ensures accuracy in solving for the unknown variable. Understanding proportions and their applications in real-world scenarios, such as in recipes or scale models, is essential for mathematical fluency.
Key Concepts
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Proportions: Equations that state two ratios are equal.
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Cross-Multiplication: A technique to solve for an unknown in proportions.
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Ratios: Relationships between two or more quantities.
Examples & Applications
Example 1: If \( \frac{2}{3} = \frac{x}{12} \), then cross-multiplication yields \( 2 \times 12 = 3 \times x \), leading to \( 24 = 3x \) and subsequently \( x = 8 \).
Example 2: For \( \frac{5}{x} = \frac{15}{9} \), we cross-multiply to get \( 5 \times 9 = 15 \times x \), resulting in \( 45 = 15x \) and thus \( x = 3 \).
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
If two ratios are set to see, cross multiply to solve with glee.
Stories
Imagine youβre baking. When a friend says double the recipe, you use a proportion to see how much flour to use! Thatβs cross-multiplication in action!
Memory Tools
Use the acronym CROSS - Calculate ratios, Repeat diagonally, Output the values, Solve for unknowns.
Acronyms
Remember MARC**
M**ultiply
**A**cross
**R**eplace
**C**onclude!
Flash Cards
Glossary
- Proportion
An equation that expresses the equality of two ratios.
- CrossMultiplication
A method used to solve proportions by multiplying the numerator of one ratio by the denominator of the other.
- Ratio
A relationship between two quantities, expressing how many times one value contains or is contained within the other.
- Unknown Variable
A symbol in an equation that represents a value that needs to be determined.
Reference links
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