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Chapter 5: Problem 40

Solve. $$ \frac{3}{y}=\frac{5}{4.5} $$

Short Answer

Expert verified
The value of \( y \) is 2.7.

Step by step solution

01

Understand the Given Equation

The given equation is: \[ \frac{3}{y} = \frac{5}{4.5} \]This is a proportion, where the two fractions are equal.
02

Simplify the Right-Hand Side

First, simplify the fraction on the right-hand side: \[ \frac{5}{4.5} = \frac{5}{\frac{9}{2}} \]Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{5}{\frac{9}{2}} = 5 \times \frac{2}{9} = \frac{10}{9} \]Therefore, the equation becomes: \[ \frac{3}{y} = \frac{10}{9} \]
03

Cross-Multiply

To solve for \( y \), cross-multiply the fractions: \[ 3 \times 9 = 10 \times y \]This simplifies to: \[ 27 = 10y \]
04

Solve for \( y \)

Isolate \( y \) by dividing both sides of the equation by 10: \[ y = \frac{27}{10} = 2.7 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
When we simplify fractions, we reduce them to their simplest form. This means making the numerator and the denominator as small as possible while keeping the value the same.
Take the fraction on the right-hand side of the given equation: \( \frac{5}{4.5} \). First, we need to recognize that dividing by a fraction is the same as multiplying by its reciprocal:
\( \frac{5}{4.5} = \frac{5}{\frac{9}{2}} = 5 \times \frac{2}{9} = \frac{10}{9} \).
Simplifying fractions makes equations easier to handle:
  • Find the reciprocal of the denominator.
  • Multiply the numerator by this reciprocal.
  • Simplify the result if possible.

By simplifying to \( \frac{10}{9} \), the right-hand side becomes manageable.
Cross Multiplication
Cross multiplication is a method used to solve proportions, where two fractions are set equal to each other.
It's very handy for solving equations like: \( \frac{3}{y} = \frac{10}{9} \).
To cross-multiply, follow these steps:
  • Multiply the numerator of the first fraction by the denominator of the second fraction.
  • Multiply the denominator of the first fraction by the numerator of the second fraction.

This looks like:
\( 3 \times 9 = 10 \times y \)
This way, we moved from a proportion to a simple equation: \( 27 = 10y \). Cross multiplication is helpful because it eliminates the fractions, making the equation easier to solve.
Solving for Variables
To solve for variables means to isolate the variable on one side of the equation.
In our example, we need to solve for \( y \) in the equation:
\( 27 = 10y \). We want \( y \) alone on one side.
To do this:
  • Divide both sides of the equation by the coefficient of \( y \) (which is 10 here).

This gives us:
\( y = \frac{27}{10} = 2.7 \).
Remember:
  • Identifying the coefficient is crucial.
  • Perform the same operation on both sides to maintain the equality.

Solving for variables can initially seem tricky, but with practice and careful steps, it becomes straightforward.

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