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Chapter 6: Problem 418

In the following exercises, translate to a proportion. What percent of \(1,000\) is 15?

Short Answer

Expert verified
1.5%

Step by step solution

01

Understand the Problem

The problem is asking what percent 15 is of 1,000. We need to translate this into a proportion.
02

Set Up the Proportion

Let the percentage be represented by the variable \(x\). Since the percentage means per 100, we create the proportion: \(\frac{x}{100} = \frac{15}{1000}\).
03

Cross Multiply to Solve for x

To solve the proportion \( \frac{x}{100} = \frac{15}{1000} \), cross multiply: \( x \times 1000 = 15 \times 100 \)
04

Simplify the Equation

After cross multiplying, we get: \ 1000x = 1500 \. Divide both sides by 1000 to solve for \(x\): \ x = \frac{1500}{1000} \
05

Simplify the Fraction

Simplify the fraction \( \frac{1500}{1000} \) to get: \ x = 1.5 \.

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Key Concepts

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

Percentage Calculation
Understanding percentage calculations is fundamental for solving many real-world problems. Percentages are a way to express a number as a part of 100. To find what percent one number is of another:
  • First, identify the part and the whole. In our exercise, 15 is the part, and 1,000 is the whole.
  • Next, set up the proportion. We use the variable x to represent the unknown percentage. This translates to: \(\frac{x}{100} = \frac{15}{1000}\).
By solving this proportion, you can find the value of \(x\), which tells you what percent 15 is of 1,000.
Cross Multiplication
Cross multiplication is a powerful technique for solving equations involving fractions. When given a proportion like \(\frac{a}{b} = \frac{c}{d}\), you can cross multiply to find the unknown value. This means:
  • Multiply the numerator of the first fraction by the denominator of the second fraction and
  • Multiply the denominator of the first fraction by the numerator of the second fraction.
In our exercise, cross multiplying \(\frac{x}{100} = \frac{15}{1000}\) results in:
\[x \times 1000 = 15 \times 100\]
This simplifies to \(1000x = 1500\). By dividing both sides of the equation by 1,000, we can isolate x and find its value.
Simplifying Fractions
Simplifying fractions is essential for making calculations easier and more understandable. To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD).
Let's consider the fraction from our exercise: \(\frac{1500}{1000}\). To simplify it:
  • Find the GCD of 1,500 and 1,000, which is 500.
  • Divide both the numerator and the denominator by 500.
This gives us:
\(\frac{1500 \div 500}{1000 \div 500} = \frac{3}{2}\).
Since we are looking for a percentage, convert \(\frac{3}{2}\) back to a decimal by performing the division \(\frac{1500}{1000} = 1.5\).
Thus, 15 is 1.5% of 1,000.

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