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Chapter 6: Problem 29

Translate to a proportion. Do not solve. $$ 80 \% \text { of what is } 16 ? $$

Short Answer

Expert verified
Set up the proportion: \(\frac{80}{100} = \frac{16}{x}\)

Step by step solution

01

- Identify the percentage

Identify the percentage given in the problem. Here, the percentage is 80%. In a mathematical context, 80% can be written as the fraction \(\frac{80}{100}\) or the decimal 0.80.
02

- Identify the part

Identify the part mentioned in the problem. The problem states that the part is 16 (80% of what is 16?).
03

- Let the whole be represented by a variable

When there's an unknown quantity in a problem, represent it with a variable. Let this unknown whole be represented by the variable \(x\).
04

- Set up the proportion

Set up a proportion equation representing the relationship between the part, the whole, and the percentage. Given that the part is 16 and the percentage is 80%, the equation is set up as: \[\frac{80}{100} = \frac{16}{x}\]

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Key Concepts

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

proportion
A proportion is a statement that two ratios are equal. In the context of our problem, we're dealing with an equation that relates parts and wholes. Proportions are often used to solve problems involving percentages or scaling. The key idea is that the relationship between the numerator and denominator of one fraction is the same as the relationship in the other fraction. The format of a proportion is: \text{{Fraction 1}} = \text{{Fraction 2}}. For example, in our problem we use the fractions \[\frac{80}{100}\] and \[\frac{16}{x}\] to set up the proportion.
percentage
A percentage represents a fraction of 100. It's used to describe proportions, comparing the part to the whole. In our problem, 80% means 80 out of every 100 units. In mathematical expressions, percentages can be converted to decimals or fractions to simplify calculations. For example, 80% can be written as \(\frac{80}{100}\) or 0.80. Thereby, understanding percentages is crucial for setting up proportions and solving related problems.
variable representation
A variable in mathematics is a symbol used to represent an unknown value. It's often denoted by letters like x or y. In the context of our problem, the unknown whole quantity is represented by the variable \(x\). Doing so helps us create an equation that we can solve. Using variables allows us to generalize problems and find solutions systematically: \text{{portion}} = \text{{percentage}} \times \text{{whole}}.
equation setup
Setting up an equation involves translating a word problem into a mathematical statement using variables and constants. In our exercise, we aim to create an expression that relates the part (16), the percentage (80%), and the whole (x). Following these steps:
  • Express the percentage as a fraction \( \frac{80}{100} \).
  • Recognize the given part is 16.
  • Let the unknown whole be x.
  • Formulate the proportion: \( \frac{80}{100} = \frac{16}{x} \).
This sets up a useful mathematical relationship that can be solved to find the unknown value represented by x.

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