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Operations with Parts

Mastering Addition, Subtraction, Multiplication, and Division of Fractions & Decimals

We know how to perform operations on whole numbers: 2 + 2 is easy. But what happens when we need to add half a cup of flour to a third of a cup? Or multiply $12.50 by 0.5? Performing operations on fractions and decimals requires a special set of rules. These rules ensure that we are comparing "apples to apples" and not mixing up parts of different sizes.

1. Operations with Fractions

Fractions are notoriously tricky because the rules change depending on which operation you are using.

Addition and Subtraction

The Golden Rule: You MUST have a Common Denominator.

You cannot add 1/2 (a half pizza) and 1/3 (a third pizza) and say you have 2/5. The slices are different sizes!

[Image of adding fractions with unlike denominators]
  1. Find the Least Common Multiple (LCM) of the denominators. For 2 and 3, it is 6.
  2. Convert the fractions: 1/2 becomes 3/6, and 1/3 becomes 2/6.
  3. Add the numerators only: 3 + 2 = 5.
  4. Keep the denominator the same: The result is 5/6.

Multiplication

This is actually the easiest operation for fractions.

The Rule: Multiply straight across.

(2/3) x (4/5)
Top x Top = 2 x 4 = 8
Bottom x Bottom = 3 x 5 = 15
Result = 8/15

Division

The Rule: Keep, Change, Flip.

Dividing by a fraction is the same as multiplying by its reciprocal.

[Image of dividing fractions keep change flip]
(1/2) / (1/4)
1. Keep the first fraction (1/2)
2. Change division to multiplication (x)
3. Flip the second fraction (4/1)
Result: (1/2) x (4/1) = 4/2 = 2

2. Operations with Decimals

Decimals are often friendlier than fractions, but they care deeply about where you put the decimal point.

Addition and Subtraction

The Golden Rule: Line up the dots.

When adding decimals, you must ensure that you are adding tenths to tenths and hundredths to hundredths.

12.5
+ 3.42
-------

Add a "ghost zero" to fill the gap: 12.50 + 3.42 = 15.92.

Multiplication

The Rule: Ignore the dot, then count.

  1. Multiply the numbers as if they were whole numbers. (0.5 x 0.3 becomes 5 x 3 = 15).
  2. Count the total number of digits behind decimal points in the original problem. (0.5 has one, 0.3 has one. Total = 2).
  3. Move the decimal point that many spaces from the right in your answer. (15 becomes 0.15).

Division

The Rule: Move the wall.

You cannot divide by a decimal. If you need to calculate 10 / 0.5, you must turn the divisor (0.5) into a whole number.

  1. Move the decimal in 0.5 one step to the right to make it 5.
  2. To keep it fair, you must also move the decimal in 10 one step to the right, making it 100.
  3. Now divide: 100 / 5 = 20.

3. Mixing Operations

Sometimes you have a fraction and a decimal in the same problem. For example: 1/2 + 0.25.

You have two choices:

  • Go Decimal: Turn 1/2 into 0.5. Then add 0.5 + 0.25 = 0.75.
  • Go Fraction: Turn 0.25 into 1/4. Then add 1/2 + 1/4. Find a common denominator (4) -> 2/4 + 1/4 = 3/4.

Both answers (0.75 and 3/4) are correct because they represent the same value.

4. Conclusion

Operations with fractions and decimals allow us to work with the real world, which is rarely composed of whole numbers. Whether you are splitting a bill, measuring wood for a project, or calculating a recipe, understanding how to add, subtract, multiply, and divide these parts is a critical life skill.