Math Problem Statement

How could you write 60/75 as a decimal without using long division? Show your work and explain your thinking

Solution

You can write 6075\frac{60}{75} as a decimal by simplifying the fraction first and then converting it to a decimal. Here's the step-by-step process:

Step 1: Simplify the fraction

To simplify 6075\frac{60}{75}, find the greatest common divisor (GCD) of 60 and 75.

  • Prime factorization of 60: 60=22×3×560 = 2^2 \times 3 \times 5
  • Prime factorization of 75: 75=3×5275 = 3 \times 5^2

The common factors are 3 and 5. So, the GCD of 60 and 75 is 1515.

Now, divide both the numerator and the denominator by their GCD (15):

6075=60÷1575÷15=45\frac{60}{75} = \frac{60 \div 15}{75 \div 15} = \frac{4}{5}

Step 2: Convert 45\frac{4}{5} to a decimal

Now, convert 45\frac{4}{5} into a decimal. We know that:

45=0.8\frac{4}{5} = 0.8

So, 6075\frac{60}{75} as a decimal is 0.8.

Explanation:

Instead of using long division, we simplified the fraction and used basic fraction-to-decimal conversion for a simpler denominator. Since 45\frac{4}{5} is a well-known fraction that equals 0.8, we can easily write the decimal.


Would you like further details or have any questions? Here are five related questions to explore:

  1. How do you simplify other fractions like 4864\frac{48}{64}?
  2. Can you convert 78\frac{7}{8} to a decimal?
  3. What is the decimal representation of 34\frac{3}{4}?
  4. How can you convert a repeating decimal to a fraction?
  5. Why is it useful to simplify fractions before converting them to decimals?

Tip: Simplifying a fraction before converting it to a decimal can make the conversion process much easier!

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Math Problem Analysis

Mathematical Concepts

Fractions
Decimal Conversion
Simplification
Greatest Common Divisor (GCD)

Formulas

Simplification: GCD of numerator and denominator
Fraction to Decimal Conversion

Theorems

None explicitly, but uses the concept of simplifying fractions

Suitable Grade Level

Grades 6-8