Problem 10 - Prealgebra

Question

Solve each proportion. $$\frac{5}{h}=\frac{10}{30}$$

Step-by-Step Solution

Answer

The value of \( h \) is 15.

Step 1: Identify the Proportion

We have the proportion \( \frac{5}{h} = \frac{10}{30} \). Our goal is to solve for \( h \). A proportion states that two ratios are equal, and we must solve for the unknown variable \( h \).

Step 2: Simplify the Known Ratio

The ratio on the right side of the equation is \( \frac{10}{30} \), which can be simplified. \( \frac{10}{30} \) is equal to \( \frac{1}{3} \). So the proportion simplifies to: \( \frac{5}{h} = \frac{1}{3} \).

Step 3: Use Cross-Multiplication

To eliminate the fractions, we use cross-multiplication. Multiply across the diagonal: \( 5 \times 3 = h \times 1 \). This results in the equation: \( 15 = h \).

Step 4: Solve for h

From the equation \( 15 = h \), we have isolated \( h \) and determined its value. Therefore, \( h = 15 \).

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing how many times one value contains or is contained within the other. They are usually expressed with a colon, like \( a:b \), or as a fraction, \( \frac{a}{b} \). In our example, we have the ratio \( \frac{5}{h} \), where 5 and \( h \) are being compared.
This tells us how 5 relates to \( h \). Ratios are powerful because they show relationships between numbers succinctly.
They can help solve problems like scaling recipes, dividing quantities, or converting measurements.

• For example, if you have a recipe that asks for a ratio of 1:2 for flour to sugar, and you have 2 cups of flour, you'll need 4 cups of sugar.
• In our exercise, the two ratios \( \frac{5}{h} \) and \( \frac{10}{30} \) are set equal to each other, creating a proportion which is essential for comparing them.

Mastering Cross-Multiplication

Cross-multiplication is a technique used to solve proportions, where you compare two ratios and solve for an unknown. It involves multiplying the terms diagonally across the equals sign.
This process eliminates the fractions, making it easier to find the value of the unknown variable.
In our example, we have the proportion \( \frac{5}{h} = \frac{1}{3} \). To clear the fractions, we multiply across the diagonals to obtain the equation \( 5 \times 3 = h \times 1 \).
This simplifies to \( 15 = h \).

• Cross-multiplication works because of the fundamental property of proportions: multiplying both sides by the same quantity maintains equality.
• It's an efficient way to handle proportions, especially when dealing with unknowns.

Simply remember, when you have an equation that equates two fractions, cross-multiplying will bring you closer to the solution.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This makes it easier to understand the fraction and its value.
For instance, in our example, the fraction \( \frac{10}{30} \) simplifies to \( \frac{1}{3} \).
This is done by dividing the numerator and denominator by their greatest common divisor, which in this case is 10.

• Simplifying helps in comparing fractions and solving problems more efficiently.
• It ensures that calculations are straightforward and less prone to error due to large numbers.

Think of simplifying as cleaning up a fraction — it reveals the most straightforward version of the quantity, making it more manageable for proportion and other mathematical operations.