# Decimals, Percentages, Mixed numbers, and improper fractions

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction, Leo Tolstoy.

# Decimals, Fractions, and Percentages

Decimals, fractions, and percentages are, basically, alternative ways of representing the same value. For instance, 0.5 = 12 = 50% could represent half of a yummy pizza; 34 inch is a fraction of the base unit (three quarters of an inch), equals to 0.75 inches, or think of it as a halfway between 12 inch and a full inch or 75%.

How do we convert between them? It’s not as difficult as it looks. Let’s convert fractions (34) to decimals. Just divide the numerator (the top part, 3) by the denominator (the bottom part, 4), so 34 = 0.75.

Next, we will convert decimals (0.75) to fractions:

• Rewrite it as a fraction: 0.751.
• Multiply the numerator and denominator by 10 for each digit to the right of the decimal point, 0.751 = 7.5 (0.75 x 10) ⁄ 10 (1 x 10) = 75 (7.5 x 10)⁄ 100 (10 x 10).
• Reduce or simplify the fraction, 75100 = 34.

Let’s convert fractions (34) to percentages. Divide the numerator (the top part, 3) by the denominator (the bottom part, 4, as 34 = 0.75), multiply the result by 100, and add the % sign, 0.75 x 100 = 75%.

How do you convert percentages (42.5%) to fractions?

• Write it down as a fraction divided by 100, 42.5% = 42.5100.
• If the percent is not a whole number, multiply the numerator and denominator by 10 for each digit to the right of the decimal point, 42.5100 = 425 (42.5 x 10)⁄ 1000 (100 x 10).
• Reduce or simplify the fraction, 4251000 = 1740.

Next, let’s convert decimals (0.25) to percentages. Multiply by 100 and add the % sign, 0.25 x 100 = 25%.

Do you want to convert percentages (45%) to decimals? Easy peasy lemon squeezy. Just divide by 100 and drop the % sign, 45 : 100 = 0.45. If you want to use Maxima to convert from fractions to decimals, you should express the numerator or denominator as a decimal by attaching a decimal point and a trailing zero to the end of it (12.0; = 0.5) or enclose the calculation within “float()” as float(34); = 0.75 Alternatively, you can use Wolfram Alpha: 35%, convert 14 to percent, 25% of 200, etc.

# Mixed numbers and improper fractions

Improper fractions and mixed numbers are basically two different ways of expressing the same number, they are both used when you need to count whole things and parts of things at the same time.

Mixed numbers, such as 2 34, 3 14, or 4 35, are made up of a whole integer (2) and a fraction (34). Think of it as we are counting whole things (2 yummy pizzas), and parts of things (three-quarters of a third pizza) at the same time.

Let’s convert a mixed number (2 34) to an improper fraction. First, multiply the whole number by the denominator and add the numerator: 2 * 4 + 3 = 11. This is the new numerator. Then, use the denominator of the fraction part of the mixed number as the denominator, 2 34= 114.

As you can see in WolframAlpha, an improper fraction, such as 85, 74 or 32 has a numerator (8) greater than (or equal to) the denominator (5).

It corresponds to 1 35 and you can read it as one pizza and three-fifths of a second one. It also means eight pieces of two pizzas that have been divided into five pieces, so you have a whole pizza -5 pieces- and three pieces of the second. Qalculate! is a multi-purpose cross-platform desktop calculator. It is simple to use but provides power and versatility normally reserved for complicated math packages, as well as useful tools for everyday needs (such as currency conversion and percent calculation). The top buttons (from left to right) switches between the general keypad and the programming keypad, switches between exact and approximate calculation, changes rational number form (it is selected Mixed fractions), selects display mode and selects number base in result

If you want to convert an improper fraction 73 to a mixed number, divide the numerator by the denominator (7 : 3 = 2 with a reminder of 1). Write down the quotient 2. Next, write down a fraction with the remainder (1) as the numerator and the same denominator of the improper fraction (3), 73 = 213.

Another program that you can use to learn fractions is KBruch (Ubuntu, Debian and derived distributions: sudo apt install kbruch). It is a small program to practice calculating with fractions and percentages.

Different exercises are provided for this purpose: arithmetic, comparison (you have to compare between two given fractions using the symbols >, < or =), conversion (you are given the task to convert a number into a fraction), factorization (in this exercise you have to factorize a given number into its prime factors), mixed numbers, and percentage. It is part of The KDE Education Project, a free educational suite for children and adults, students and teachers. It is available in GNU/Linux and Windows. In our article Fractions in Python, you can learn how to work with fractions in Python.

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