Question

Divide the following numbers, and round to the nearest hundredth. \(0.13 \div 0.25=\) ______

Short answer

0.13 divided by 0.25 is 0.52.

Step-by-step solution

Set Up the Division Problem

Write down the division problem as a fraction: \( \frac{0.13}{0.25} \). This allows us to focus on solving the problem step by step.

Convert Decimals to Fractions

Convert each decimal to a fraction. \(0.13\) is \(\frac{13}{100}\) and \(0.25\) is \(\frac{25}{100}\). Now the division problem looks like:\[ \frac{13}{100} \div \frac{25}{100} \]

Perform the Division of Fractions

To divide fractions, multiply by the reciprocal. The division \( \frac{13}{100} \div \frac{25}{100} \) becomes \( \frac{13}{100} \times \frac{100}{25} \).

Simplify the Multiplication

Multiply the numerators and the denominators:\[ \frac{13 \times 100}{100 \times 25} = \frac{1300}{2500} \]

Simplify the Fraction

Simplify \( \frac{1300}{2500} \) by finding the greatest common divisor (GCD), which is 100. This simplifies to:\[ \frac{1300 \div 100}{2500 \div 100} = \frac{13}{25} \].

Convert the Simplified Fraction to Decimal

Convert \( \frac{13}{25} \) to a decimal by dividing 13 by 25:\[ 13 \div 25 = 0.52 \]

Round to the Nearest Hundredth

Check the decimal place values. Since it already is 0.52, rounding is not necessary.

Key concepts

Rounding Numbers

Rounding numbers is a valuable skill, especially when working with decimals. It involves reducing the number of digits in a number while trying to keep its value close to the original. The goal is to simplify and make it easier to work with.
When rounding to the nearest hundredth, focus on the second decimal place. Look at the third decimal place to decide whether to round up or keep the same:

• If the third decimal place is 5 or more, increase the second decimal place by 1.
• If it is less than 5, leave the second decimal place unchanged.

In our exercise, the result was already at 0.52 after conversion, so no rounding was necessary. This step ensures that the number is precise to two decimal places, which is helpful for various calculations.

Converting Decimals to Fractions

Converting decimals into fractions is a useful way to handle division problems, especially when the numbers involved are not whole numbers.
To convert a decimal to a fraction:

• Write down the decimal divided by 1. For example, 0.13 becomes \( \frac{0.13}{1} \).
• Multiply both the numerator and the denominator by 10 for each decimal place to eliminate the decimals. For 0.13, since it has two decimal places, multiply by 100 to get \( \frac{13}{100} \).

This converts the decimal into a fraction that can be easily manipulated. Converting decimals to fractions is crucial as it allows the use of fraction properties in mathematical operations.

Simplifying Fractions

Simplifying fractions is an essential concept to reduce fractions to their simplest form. This process involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.
In our example, after setting up the division and performing multiplication, we had \( \frac{1300}{2500} \). To simplify:

• Find the GCD of 1300 and 2500, which is 100.
• Divide both the numerator and the denominator by the GCD: \( \frac{1300 \div 100}{2500 \div 100} = \frac{13}{25} \).

Simplifying fractions makes calculations easier and answers cleaner. It helps in getting results that are neat and understandable.

Decimal to Fraction Conversion

Decimal to fraction conversion is a slightly different yet closely related process to converting decimals to fractions. It focuses on expressing a decimal number as a fraction more naturally.
The decimal 0.13 is directly written as \( \frac{13}{100} \), which involves considering the place value of the decimal:

• For each digit after the decimal, count the position as powers of ten. The hundredth place makes it \( 13 \) out of \( 100 \).

This conversion is beneficial when wanting to see numbers in a fraction form, which may be more intuitive in some contexts. Understanding this concept is vital in interpreting fractions and decimals interchangeably.