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Chapter 0: Problem 2

By using a calculator, or otherwise, simplify the following: a \(1 \frac{1}{5}: 2 \frac{1}{4}\) b \(3 \frac{1}{2}: 2 \frac{5}{12}\)

Short Answer

Expert verified
a) 8/15, b) 42/29

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

For part (a), convert the mixed fractions to improper fractions. The fraction mixed number for 1 1/5 is: 1 1/5 = 1 + 1/5 = 6/5And for 2 1/4: 2 1/4 = 2 + 1/4 = 9/4So the problem becomes 6/5 divided by 9/4.
02

Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.Part (a): (6/5) 梅 (9/4) = (6/5) * (4/9)Multiply the numerators: 6 * 4 = 24Multiply the denominators: 5 * 9 = 45So the result is 24/45.
03

Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 24 and 45, which is 3.24 梅 3 = 845 梅 3 = 15So, the simplified fraction is 8/15.
04

Repeat Steps for Part (b)

Convert the mixed fractions to improper fractions. For 3 1/2:3 1/2 = 3 + 1/2 = 7/2And for 2 5/12:2 5/12 = 2 * 12/12 + 5/12 = 24/12 + 5/12 = 29/12So the problem becomes 7/2 divided by 29/12.
05

Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.Part (b): (7/2) 梅 (29/12) = (7/2) * (12/29)Multiply the numerators: 7 * 12 = 84Multiply the denominators: 2 * 29 = 58So the result is 84/58.
06

Simplify the Fraction

Simplify the fraction by finding the GCD of 84 and 58, which is 2.84 梅 2 = 4258 梅 2 = 29So, the simplified fraction is 42/29.

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Key Concepts

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

mixed numbers
Mixed numbers are numbers that contain both a whole number and a fraction. They are often used in everyday measurements, like cooking or construction. For example, if you have a piece of wood that is 3 1/2 feet long, that is a mixed number. To work with mixed numbers in mathematical operations, you often need to convert them to improper fractions.
  • A mixed number like 3 1/2 can be converted by multiplying the whole number by the denominator of the fraction and then adding the numerator.
  • For 3 1/2, multiply 3 by 2 to get 6, and then add 1 to get 7. So, 3 1/2 becomes 7/2.
Mixed numbers make it easier to understand and visualize quantities, but using improper fractions simplifies calculations.
improper fractions
Improper fractions are fractions where the numerator is greater than or equal to the denominator. For example, 7/4 and 9/8 are improper fractions. They are particularly useful in mathematical calculations because they simplify the arithmetic processes.
  • To convert a mixed number to an improper fraction, multiply the whole number by the denominator and then add the numerator.
  • For example, 1 1/5 becomes 6/5 because 1*5 + 1 equals 6, over the original denominator 5.
Improper fractions can also be converted back to mixed numbers for better visualization, simply by dividing the numerator by the denominator and expressing the remainder as a fraction. For instance, 7/2 can be written as 3 1/2.
greatest common divisor
The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It is crucial when simplifying fractions.
  • For example, to simplify the fraction 24/45, you need to find the GCD of 24 and 45, which is 3.
  • Divide both the numerator and denominator by their GCD: 24 梅 3 = 8 and 45 梅 3 = 15.
  • So, the simplified fraction is 8/15.
Knowing how to find the GCD helps in reducing fractions to their simplest form, making them easier to work with and understand.
You can find the GCD using the Euclidean algorithm, which involves dividing and taking remainders repeatedly until you reach a remainder of 0. The divisor at this point is the GCD.

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Most popular questions from this chapter

The exact value of sound velocity is \(342 \mathrm{~m} / \mathrm{s}\). If the experimental value produces \(345 \mathrm{~m} / \mathrm{s}\), find the percentage error.

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