Problem 24 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{n}{10}=\frac{\frac{3}{5}}{\frac{3}{8}}$$

Step-by-Step Solution

Answer

The missing term \( n \) is 16.

Step 1: Set Up the Proportion Equation

We are given the proportion equation \( \frac{n}{10} = \frac{\frac{3}{5}}{\frac{3}{8}} \). The task is to find the value of \( n \) that completes this proportion.

Step 2: Simplify the Complex Fraction

To simplify \( \frac{\frac{3}{5}}{\frac{3}{8}} \), perform the division by converting it to a multiplication of fractions: \( \frac{3}{5} \div \frac{3}{8} = \frac{3}{5} \times \frac{8}{3} \).

Step 3: Multiply the Fractions

Multiply the simplified fractions: \( \frac{3}{5} \times \frac{8}{3} = \frac{3 \times 8}{5 \times 3} = \frac{24}{15} \).

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{24}{15} \) by finding the greatest common divisor (GCD) of 24 and 15, which is 3. Divide the numerator and denominator by 3: \( \frac{24}{15} = \frac{8}{5} \).

Step 5: Equate and Cross-Multiply

Now the equation is \( \frac{n}{10} = \frac{8}{5} \). Cross-multiply to solve for \( n \): \( n \times 5 = 8 \times 10 \).

Step 6: Solve for n

Simplify and solve for \( n \): \( 5n = 80 \). Divide both sides by 5: \( n = \frac{80}{5} = 16 \).

Key Concepts

Fraction Simplification

Fraction simplification is like tidying up a fraction to make it as simple as possible. Imagine you have a fraction with both a top number (numerator) and a bottom number (denominator) that have a common factor, meaning they can both be divided by the same number. When you simplify a fraction, you divide both by their greatest common divisor (GCD). This means the fraction looks cleaner, but it still represents the same value.
For example, if you have \( \frac{24}{15} \), you simplify it by finding the GCD of 24 and 15, which is 3. Divide both 24 by 3 and 15 by 3 to simplify it to \( \frac{8}{5} \). Simplifying fractions makes them easier to work with and compare to other fractions.

Cross Multiplication

Cross multiplication is a handy method for solving equations that involve two ratios or fractions set equal to each other. Think of it as a way to clear the fractions by multiplying in a cross pattern.
In the equation \( \frac{n}{10} = \frac{8}{5} \), cross multiplication means you multiply the numerator of each fraction by the denominator of the other:

• The 5 is multiplied by \( n \) (\( 5 \times n \)).
• The 8 is multiplied by 10 (\( 8 \times 10 \)).

So, you would write the equation as \( 5n = 80 \). By cross multiplying, you've eliminated the fractions, making it easier to solve for the unknown \( n \). When you divide both sides by 5, you find that \( n = 16 \). It's like magic for solving proportion equations!

Complex Fractions

Complex fractions are like fractions in a fraction sandwich! They have a fraction in either the numerator, the denominator, or both. To simplify complex fractions, convert the division of fractions into multiplication.
Take, for instance, \( \frac{\frac{3}{5}}{\frac{3}{8}} \). It looks complicated at first, but you can untangle it by turning the division into multiplication. To do this, you take the reciprocal (flip) of the bottom fraction and multiply. This means \( \frac{3}{5} \div \frac{3}{8} \) becomes \( \frac{3}{5} \times \frac{8}{3} \).
The result is \( \frac{24}{15} \), which you can then simplify. By breaking down complex fractions this way, you make them much easier to handle and solve.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial tool for simplifying fractions. It's like a secret key that unlocks the simplest form of a fraction. The GCD is the largest whole number that can evenly divide both the numerator and the denominator of a fraction.
For example, in the fraction \( \frac{24}{15} \), both 24 and 15 can be divided by 3, the largest number that fits this description for both. Figuring out the GCD involves checking the numbers for common factors, and 3 is the highest one in this case. By dividing both parts of the fraction by the GCD, you achieve the simplest form, such as \( \frac{8}{5} \). Using the GCD ensures the fraction is as straightforward as possible.