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Chapter 7: Problem 5

Translate each to a proportion. Do not solve. 14.3 is \(26 \%\) of what number?

Short Answer

Expert verified
The proportion is \(\frac{14.3}{x} = \frac{26}{100}\).

Step by step solution

01

Identify the known values

In the exercise, we know the value 14.3 and that it represents 26% of some unknown number. We start by identifying these values.
02

Set up the proportion

To form a proportion, equate the known percentage to a fraction. Write 26% as \(\frac{26}{100}\), and use this to set up the equivalent proportion with the known number 14.3 over an unknown, which we will call \(x\).
03

Write the proportion equation

Using the information from Step 2, the proportion is \(\frac{14.3}{x} = \frac{26}{100}\). This equation relates 14.3 as 26% of the unknown number \(x\).

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Key Concepts

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

Percentages
Percentages are a way of expressing numbers as a part of a whole. Essentially, a percentage is a fraction with a denominator of 100. When we say "26%", we mean 26 out of 100 total parts.

Using percentages is helpful in understanding parts of a given whole, like determining what percent of a test score you achieved. To work with percentages mathematically, it's crucial to be comfortable converting them into fractions or decimals. For example:
  • 26% can be written as the fraction \( \frac{26}{100} \).
  • Alternatively, it can be converted to a decimal as 0.26.
These conversions are instrumental when setting up proportions or solving other math problems involving percentages.
Fractions
Fractions are a way to represent a part of a whole, where we have a numerator above and a denominator below a dividing line. The denominator shows how many equal parts the whole is divided into, while the numerator indicates how many of those parts we are considering.

In our exercise, we converted a percentage into a fraction (i.e., \( \frac{26}{100} \)). This fraction form is central when setting up proportions, as it helps compare the fractional part of one quantity to another. Fractions are versatile and can be used in
  • Adding or subtracting quantities.
  • Comparing and measuring different parts of a whole.
  • Solving proportions where fractions represent relationships between numbers.
Understanding fractions is key to solving problems involving parts of a whole, just like percentages.
Unknown Variables
Unknown variables are symbols, often denoted by letters (like \(x\)), representing a quantity we do not yet know or are trying to find. In mathematics, they are placeholders for the numbers we want to find. Variables are commonly used in equations to express general formulas or relationships between numbers.

In our given problem, \(x\) is the unknown variable representing a number we aim to discover so that 14.3 is 26% of it. This use of variables allows us to create an equation, like a proportion, to solve for the unknown quantity.
  • Variables can be part of simple equations like linear equations.
  • They can express unknown quantities in formulas, making substitution easier once we find the variable's value.
  • Using variables simplifies the representation of unknowns in problems, adding clarity and rigor to mathematical expressions.
Mastering the use of unknown variables is essential, as they form the basis of algebraic thinking and problem solving.

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