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Chapter 5: Problem 32

Find the ratio of the first number to the second and simplify. $$ 40 \text { to } 60 $$

Short Answer

Expert verified
The simplified ratio is 2 to 3.

Step by step solution

01

- Write the Ratio as a Fraction

Begin by expressing the ratio of 40 to 60 as a fraction: \[ \frac{40}{60} \]
02

- Find the Greatest Common Divisor (GCD)

Find the greatest common divisor of 40 and 60. The GCD of 40 and 60 is 20.
03

- Simplify the Fraction

Divide both the numerator and the denominator by their GCD (20): \[ \frac{40 \text{ 梅 } 20}{60 \text{ 梅 } 20} = \frac{2}{3} \]
04

- Express the Simplified Ratio

Write the simplified ratio. The ratio of 40 to 60 simplified is 2 to 3.

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Key Concepts

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

Ratios
A ratio is a way to compare two quantities by showing the relative size of one quantity to the other. Ratios can be written in different ways, such as 'a to b,' 'a:b,' or as a fraction \(\frac{a}{b}\).
  • **Example:** If you have 10 apples and 5 oranges, the ratio of apples to oranges is 10 to 5.
Ratios help us understand relationships between numbers and can be used in various fields such as cooking, science, and finance.
Simplifying ratios makes them easier to work with and understand.
Greatest Common Divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), of two numbers is the largest number that divides both of them without leaving a remainder.
Finding the GCD is a crucial step in simplifying fractions.
**How to find the GCD:**
  • List all the factors of each number.
  • Identify the largest factor that both numbers have in common.

**Example:** For the numbers 40 and 60:
  • Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • Common factors: 1, 2, 4, 5, 10, 20
  • GCD: 20
Using the GCD, we can simplify fractions more efficiently.
Fraction Simplification
Simplifying a fraction means making it as simple as possible. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).

**Example:** Simplify the fraction \(\frac{40}{60}\).
  • Find the GCD of 40 and 60, which is 20.
  • Divide both the numerator and the denominator by the GCD: \(\frac{40 \text{ 梅 } 20}{60 \text{ 梅 } 20} = \frac{2}{3}\).
  • So, \(\frac{40}{60}\) simplifies to \(\frac{2}{3}\).

By simplifying fractions, we make ratios easier to understand and work with.

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Most popular questions from this chapter

Find the length \(L\) of the lake. Assume that the ratio of \(L\) to 120 yd is the same as the ratio of 720 yd to 30 yd.

Solve. $$ \frac{7}{\frac{1}{4}}=\frac{28}{x} $$

Solve. $$ \frac{2}{5}=\frac{12}{x} $$

Divide. Simplify, if possible. $$ 4000 \div 32 $$

Solve. $$ \frac{t}{0.16}=\frac{0.15}{0.40} $$
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