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Chapter 5: Problem 34

Find the ratio of the first number to the second and simplify. $$ 42 \text { to } 100 $$

Short Answer

Expert verified
\[\frac{21}{50}\]

Step by step solution

01

- Write the Ratio

Express the given numbers as a ratio in fraction form. Thus, the ratio of 42 to 100 is written as: \[\frac{42}{100}\]
02

- Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of 42 and 100. The GCD of 42 and 100 can be found using the Euclidean algorithm or by listing the factors:- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42- Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100The greatest common divisor is 2.
03

- Simplify the Ratio

Divide both the numerator and the denominator of the fraction by their GCD (2): \[\frac{42 \div 2}{100 \div 2} = \frac{21}{50}\]

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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. It shows how many times one number contains another. For example, in the ratio 42 to 100, we want to express how 42 compares to 100. Ratios can be written in three different forms: 42:100, 42 to 100, or \(\frac{42}{100}\). Using fraction form is very common because fractions are easy to simplify. Simplifying ratios helps to understand the relationship between two numbers better. Let鈥檚 walk through how to simplify ratios!
greatest common divisor
To simplify a ratio, we need to find the greatest common divisor (GCD) of the numbers involved. The GCD is the largest number that divides both numbers without leaving a remainder. To find the GCD, we can use the Euclidean algorithm or list the factors of each number. For the numbers 42 and 100, the factors are:
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The greatest common divisor of 42 and 100 is 2. This number helps simplify the ratio effectively.
fraction simplification
Fraction simplification involves reducing a fraction to its simplest form. To do this, we divide both the numerator and the denominator by their GCD. For the ratio \(\frac{42}{100}\), simplify it by dividing both the numerator and the denominator by 2 (the GCD). This results in:
\(\frac{42 \div 2}{100 \div 2} = \frac{21}{50}\)
So, the simplified ratio of 42 to 100 is 21 to 50. Simplifying a fraction helps in better understanding and comparing ratios.

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Most popular questions from this chapter

Multiply. Simplify, if possible. $$ 4 \frac{1}{10} \cdot 3 \frac{1}{3} $$

Solve. $$ \frac{0.12}{0.04}=\frac{t}{0.32} $$

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Find each unit price Then, in each exercise, determine which size is the better buy based on unit price alone. $$ \begin{array}{|c|c|c|} \hline \text { Size } & \text { Price } & \text { Unit Price } \\ \hline 10.7 \mathrm{oz} & \$ 1.09 & \\ \hline 26 \mathrm{oz} & \$ 2.69 & \\ \hline \end{array} $$

Solve. $$ \frac{5 \frac{1}{5}}{6 \frac{1}{6}}=\frac{y}{3 \frac{1}{2}} $$
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