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Chapter 5: Problem 3

Write ratio as a fraction and simplify. \(8 \mathrm{cm}\) to \(32 \mathrm{cm}\)

Short Answer

Expert verified
The simplified ratio is \(\frac{1}{4} \).

Step by step solution

01

- Write the Given Ratio

Write the given ratio as a fraction: \( \frac{8 \text{ cm}}{32 \text{ cm}} \)
02

- Simplify the Fraction

Find the greatest common divisor (GCD) of 8 and 32, which is 8. Divide both the numerator and the denominator by their GCD: \( \frac{8 \div 8}{32 \div 8} = \frac{1}{4} \)

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Key Concepts

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

Ratios
Ratios are a way to compare two quantities by showing how many times one value contains the other. They can be represented in various forms such as:
  • a fraction like \( \frac{3}{4} \)
  • using a colon like 3:4
  • or using the word 'to' like 3 to 4
Understanding ratios is fundamental because they appear in various mathematical contexts, from geometry to real-life scenarios like recipe adjustments and map reading.
To simplify a ratio such as 8 cm to 32 cm, you first write it as a fraction: \( \frac{8}{32} \) This makes it easier to simplify by using another core concept: fractions.
Fractions
Fractions represent a part of a whole and consist of two parts: a numerator (top number) and a denominator (bottom number). In the fraction \( \frac{8}{32} \), 8 is the numerator and 32 is the denominator.
Simplifying fractions involves reducing them to their simplest form. This is done by finding a number that can divide both the numerator and the denominator without leaving a remainder. In our example, we need to find the greatest number that can divide both 8 and 32. This leads us to our next important concept: the greatest common divisor.
Greatest Common Divisor
The Greatest Common Divisor (GCD) of two numbers is the largest positive integer that divides both numbers exactly. Finding the GCD helps in simplifying ratios and fractions correctly.
For our fraction \( \frac{8}{32} \), we need to find the GCD of 8 and 32. Breaking down the steps:
  • Factors of 8 are 1, 2, 4, 8
  • Factors of 32 are 1, 2, 4, 8, 16, 32
  • The greatest common factor is 8
This means we can divide both the numerator and the denominator of \( \frac{8}{32} \) by 8. Doing this, \( \frac{8 \div 8}{32 \div 8} = \frac{1}{4} \).
Therefore, the simplified form of the ratio 8 cm to 32 cm is \( \frac{1}{4} \), or simply 1:4 when written as a ratio.

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