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Magazine Chapter 6: Problem 44
Solve each proportion for the given variable. Round the solution where indicated. $$ \frac{12}{10}=\frac{z}{16} $$
Short Answer
Expert verified
The value of \( z \) is 19.2.
Step by step solution
01
Understand the Proportion
The given proportion is \( \frac{12}{10} = \frac{z}{16} \). It is an equation stating that two ratios are equal. Our goal is to solve for the variable \( z \).
02
Apply Cross-Multiplication
We solve the proportion by cross-multiplying. This means multiplying the numerator of one fraction by the denominator of the other fraction (and vice versa) to set up an equation. This gives: \[ 12 \times 16 = 10 \times z \].
03
Calculate the Product
Calculate \( 12 \times 16 \). Perform the multiplication:\[ 12 \times 16 = 192 \].
04
Formulate the Equation
Substitute back into the equation:\[ 192 = 10z \].
05
Solve for z
To find \( z \), divide both sides of the equation by 10:\[ z = \frac{192}{10} \].
06
Simplify the Expression
Simplify \( \frac{192}{10} \) to get \( z = 19.2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Ratios
A ratio is a way to compare two numbers or quantities. In a ratio, we often use a colon or a fraction to show this comparison. For example, the ratio \( \frac{12}{10} \) means that 12 is compared to 10. Ratios are essential in many fields like cooking, engineering, and mathematics because they help us understand relationships between different amounts.
- Ratios can be expressed in different forms. For instance, "12 to 10" can also be written as "12:10" or "\( \frac{12}{10} \)".
- Understanding the context of a ratio is crucial: it tells us how much bigger, smaller, faster, or slower one quantity is compared to another.
- In the given problem, we are comparing \( 12 \) to \( 10 \) and \( z \) to \( 16 \) via a proportion, a statement of equality between two ratios.
Mastering Cross-Multiplication
Cross-multiplication is a useful technique used to solve proportions. When we have an equation consisting of two fractions set equal to each other, we can use cross-multiplication to find the unknown variable.
- In cross-multiplication, you multiply the numerator of one fraction by the denominator of the opposing fraction and set the products equal.
- For example, in the proportion \( \frac{12}{10} = \frac{z}{16} \), we cross-multiply to get \( 12 \times 16 = 10 \times z \).
- This method helps us eliminate the fractions, which makes it easier to solve for the unknown variable.
Steps in Solving Equations
Once you have set up an equation using cross-multiplication, the next step is to solve this equation for the unknown variable. This process involves a few straightforward algebraic operations. Below are the typical steps to solve for a variable.
- Multiply or Calculate:** First, calculate any products obtained from the cross-multiplication. In our example, \( 12 \times 16 = 192 \).
- Write the Equation:** Next, substitute the product back into the equation. We have \( 192 = 10z \).
- Isolate the Variable:** To solve for \( z \), divide both sides of the equation by the coefficient of \( z \), which in this case is 10. So, \( z = \frac{192}{10} \).
- Solution:** Lastly, perform any necessary division or simplification to find the value of the variable. Here, \( z = 19.2 \).
Crafting Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operations (like addition or multiplication) that represent a value. In our context, the expressions occur in the ratios and the resulting equation after cross-multiplication.
- An expression can be simple, like \( 12 \times 16 \), or more complex, involving multiple operations or steps.
- In solving proportions, we use expressions to structure our thought process and work towards the solution. After cross-multiplying, we had the expression \( 12 \times 16 = 10z \).
- Simplifying expressions, such as dividing \( 192 \) by \( 10 \), helps leads us to the final answer.
- Mathematical expressions are the building blocks of solving equations, balancing them to maintain equality on both sides.
