Math Foundations Unit 4: How-To Guide

Part 1: Fraction Operations

How to Add or Subtract Fractions (with different denominators)

  1. Find a common denominator by multiplying the bottom numbers together.
  2. Rewrite each fraction as an equivalent fraction with the new denominator.
  3. Add or subtract the numerators. Keep the denominator the same.
  4. Simplify the final fraction if needed.

Worked Example: \( \frac{2}{3} + \frac{1}{4} \)

Step 1 (Common Denominator): Multiply the denominators: \(3 \times 4 = 12\).

Step 2 (Rewrite Fractions):
\( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
\( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)

Step 3 (Add): \( \frac{8}{12} + \frac{3}{12} = \frac{8+3}{12} = \frac{11}{12} \)

Step 4 (Simplify): \(\frac{11}{12}\) is already in simplest form.

How to Multiply Fractions

  1. Multiply the numerators (top numbers) together.
  2. Multiply the denominators (bottom numbers) together.
  3. Simplify the resulting fraction if needed.

Worked Example: \( \frac{3}{5} \times \frac{2}{9} \)

Step 1 (Multiply Numerators): \(3 \times 2 = 6\)

Step 2 (Multiply Denominators): \(5 \times 9 = 45\)

Result: \( \frac{6}{45} \)

Step 3 (Simplify): Both 6 and 45 can be divided by 3. \( \frac{6 \div 3}{45 \div 3} = \frac{2}{15} \)

How to Divide Fractions

  1. KEEP the first fraction the same.
  2. CHANGE the division sign to a multiplication sign.
  3. FLIP the second fraction to its reciprocal.
  4. Multiply the fractions and simplify if needed.

Worked Example: \( \frac{1}{2} \div \frac{3}{4} \)

Step 1 (KEEP): \( \frac{1}{2} \)

Step 2 (CHANGE): \( \frac{1}{2} \times \)

Step 3 (FLIP): \( \frac{1}{2} \times \frac{4}{3} \)

Step 4 (Multiply): \( \frac{1 \times 4}{2 \times 3} = \frac{4}{6} \). Simplify by dividing by 2 to get \( \frac{2}{3} \).

Part 2: Decimal Operations

How to Add or Subtract Decimals

  1. Line up the numbers vertically so the decimal points are in a straight line.
  2. Add zeros as placeholders in any empty spots to the right.
  3. Add or subtract as you would with whole numbers.
  4. Bring the decimal point straight down into your answer.

Worked Example: \( 24.7 + 5.91 \)

  24.70  (add a zero)
+  5.91
-------
  30.61

How to Multiply Decimals

  1. Multiply the numbers as if they were whole numbers (ignore the decimals).
  2. Count the total number of decimal places in both numbers you multiplied.
  3. Place the decimal point in your answer so it has that total number of decimal places.

Worked Example: \( 3.77 \times 2.8 \)

   3.77  (2 decimal places)
 x  2.8  (1 decimal place)
-------
  3016
+ 7540
-------
 10.556  (3 total decimal places)

How to Divide Decimals

  1. If the divisor (the number you're dividing by) is not a whole number, move its decimal point to the right to make it one.
  2. Move the decimal point in the dividend (the number being divided) the same number of places to the right.
  3. Bring the decimal point straight up into the quotient (the answer).
  4. Divide as you would with whole numbers.

Worked Example: \( 12.45 \div 0.5 \)

Step 1 & 2: Move the decimal one place to the right in both numbers.
\( 0.5 \rightarrow 5 \)
\( 12.45 \rightarrow 124.5 \)

Step 3 & 4: The problem is now \( 124.5 \div 5 \). Divide normally.

   24.9
  ____
5|124.5
  -10
  ---
   24
  -20
  ---
    45
   -45
   ---
     0

The answer is 24.9.