Cross-Canceling Before Multiplying: Fifth Grade Mathematics

Cross-canceling simplifies shared common factors between numerators and denominators before multiplying, preventing cumbersome large numbers.

The Logic of Cross-Canceling

Because multiplication is commutative, factors in the numerators and denominators can be rearranged: (a * c) / (b * d) = (a * c) / (d * b) = (a / d) * (c / b). Any factor appearing in ANY numerator can cancel with that same factor in ANY denominator.

+--------------------------------------------------------------------------+
|                  CROSS-CANCELING BEFORE MULTIPLYING                      |
|                                                                          |
|       Problem:  (4/9) * (3/8)                                            |
|                                                                          |
|       Without Cross-Canceling:                                           |
|         (4 * 3) / (9 * 8) = 12/72 = 1/6  (Large numbers to reduce)       |
|                                                                          |
|       With Cross-Canceling:                                              |
|         - Look at diagonal pair 4 and 8: both divisible by 4.            |
|           4 becomes 1; 8 becomes 2.                                      |
|         - Look at diagonal pair 3 and 9: both divisible by 3.            |
|           3 becomes 1; 9 becomes 3.                                      |
|                                                                          |
|       New Expression: (1/3) * (1/2) = (1 * 1) / (3 * 2) = 1/6           |
|       The product is already in simplest form!                           |
+--------------------------------------------------------------------------+

Visualizing Diagonal Simplification:
      1               1
     (4)             (3)            1 * 1       1
    -----     *     -----     =    -------  =  ---
     (9)             (8)            3 * 2       6
      3               2

Rules for Safe Cross-Canceling

Always cancel one numerator factor with one denominator factor.
You can NEVER cancel two numerators with each other, nor two denominators with each other.
Divide both canceled numbers by the exact same common factor.

Exercises and Solutions

Exercise 1: Double Diagonal Cross-Canceling. Multiply (15/28) * (14/25) using cross-canceling. Show each diagonal division and compute the simplified product.

Solution 1: Inspect the diagonal pair 15 and 25: both share factor 5. Divide 15 by 5 = 3; divide 25 by 5 = 5. Inspect the diagonal pair 14 and 28: both share factor 14. Divide 14 by 14 = 1; divide 28 by 14 = 2. Rewrite the expression with simplified numbers: (3/2) * (1/5). Multiply straight across: (3 * 1) / (2 * 5) = 3/10. The product is 3/10, automatically in lowest terms.

Exercise 2: Three-Fraction Multiplication. Solve (3/4) * (8/9) * (5/6) using cross-canceling.

Solution 2: Look across all numerators (3, 8, 5) and denominators (4, 9, 6). Cancel 4 in denominator with 8 in numerator: 4 becomes 1, 8 becomes 2. Now cancel 2 in numerator with 6 in denominator: 2 becomes 1, 6 becomes 3. Now cancel 3 in numerator with 3 in denominator: both become 1. The remaining numerators are 1, 1, and 5. The remaining denominators are 1, 9, and 1. Multiply remaining terms: (1 * 1 * 5) / (1 * 9 * 1) = 5/9. The final simplified product is 5/9.