Problem 19 - 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. $$\frac{x}{2}=7$$

Step-by-Step Solution

Answer

The missing term is \( x = 14 \).

Step 1: Understand the proportion

The problem gives us a proportion \( \frac{x}{2} = 7 \). This means that the ratio of \( x \) to 2 is equal to 7.

Step 2: Solve for x by cross-multiplication

To solve for \( x \), multiply both sides by 2 to eliminate the denominator. This gives us \( x = 7 \times 2 \).

Step 3: Simplify the expression

Calculate \( 7 \times 2 \) to find the value of \( x \). We get \( x = 14 \).

Step 4: Express the answer as a fraction in lowest terms

The result is \( x = 14 \). The number 14 can be expressed as \( \frac{14}{1} \), which is already in its lowest terms.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a technique used to solve equations where two ratios are equal. Imagine you have two fractions set equal to each other, like \( \frac{a}{b} = \frac{c}{d} \). Cross-multiplication involves multiplying the numerator (top number) of each fraction by the denominator (bottom number) of the opposite fraction. For example, in the proportion \( \frac{x}{2} = 7 \), you multiply \( x \) by the denominator from the other side (which is 1 considering 7 as \( \frac{7}{1} \)). Hence you do \( x \times 1 \) and set it equal to \( 7 \times 2 \).

1. Multiply \( x \) by 1 → stays as \( x \).
2. Multiply 7 by 2 → gives you 14.

Setting these equal gives \( x = 14 \). This approach helps solve for unknown variables in proportions quickly and efficiently.

Grasping Fractions

Fractions are a way of representing parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). When dealing with proportions, as in the problem \( \frac{x}{2} = 7 \), you're relating one fraction to another. In this case, \( \frac{x}{2} \) is symbolizing a part-whole relationship where x is the part, and 2 is the whole.
To solve it, understanding how fractions operate allows you to manipulate them, like eliminating the denominator by multiplying both sides by it. Here, \( 7 \) can also be seen as \( \frac{7}{1} \), which is crucial for cross-multiplication. Moreover, after solving equations, expressing the final answer in fraction form can help in maintaining consistency. Remember, fractions need to be in their simplest form, like \( \frac{14}{1} \) in this example. There is nothing to reduce here as 14 and 1 have no common factors other than 1.

Deciphering Ratios

Ratios represent the relationship between two quantities, showing how many times one number contains another. In the context of our problem \( \frac{x}{2} = 7 \), this means the relationship between \( x \) and 2 is the same as the number 7. It tells you that for every one part of 2, there are 7 parts of \( x \).
A ratio can be expressed in different ways, including fractions, colons, or as a word statement. In solving proportions with a missing term, identifying that the two ratios are equal is the first step. This understanding allows you to set up equations that can be solved using strategies like cross-multiplication. Ratios are versatile; they help compare quantities in practical scenarios, such as in cooking or determining speeds, where maintaining a consistent relationship is essential.