Reducing before multiplying
Reducing before multiplying (also called cross-canceling) is a Grade 7 fraction shortcut from Saxon Math, Course 2 that simplifies fraction multiplication by canceling common factors in any numerator and any denominator before multiplying. For example, in (9/16) × (2/3), the 9 and 3 share a factor of 3 (9÷3=3, 3÷3=1), and 16 and 2 share a factor of 2 (16÷2=8, 2÷2=1), leaving (3/8) × (1/1) = 3/8. This technique produces smaller numbers and eliminates the need to simplify large products after multiplying.
Key Concepts
Property When multiplying fractions, you can simplify the problem by pairing any numerator with any denominator and dividing both by a common factor. This method is also known as canceling.
Examples $$\frac{9}{16} \cdot \frac{2}{3} = \frac{\stackrel{3}{\cancel{9}}}{\underset{8}{\cancel{16}}} \cdot \frac{\stackrel{1}{\cancel{2}}}{\underset{1}{\cancel{3}}} = \frac{3}{8}$$ $$\frac{8}{9} \cdot \frac{3}{10} \cdot \frac{5}{4} = \frac{\stackrel{1}{\cancel{8}}}{\underset{3}{\cancel{9}}} \cdot \frac{\stackrel{1}{\cancel{3}}}{\underset{1}{\cancel{10}}} \cdot \frac{\stackrel{1}{\cancel{5}}}{\underset{1}{\cancel{4}}} = \frac{1}{3}$$ $$\frac{5}{8} \cdot \frac{3}{10} = \frac{\stackrel{1}{\cancel{5}}}{8} \cdot \frac{3}{\underset{2}{\cancel{10}}} = \frac{3}{16}$$.
Explanation Why do the hard work after you multiply? Canceling lets you shrink the numbers before you multiply, making the whole problem simpler. It's like tidying up as you go! Find a numerator denominator pair with a common factor, reduce them, and cruise to the right answer with smaller, friendlier numbers.
Common Questions
What does reducing before multiplying mean?
It means simplifying fractions by canceling common factors across numerators and denominators before you multiply, so you work with smaller numbers throughout.
Can you cancel a numerator with any denominator, not just its own fraction's denominator?
Yes. When multiplying fractions, any numerator in the expression can be canceled with any denominator. You are not restricted to within a single fraction.
How does reducing before multiplying save time?
By canceling factors early, you avoid multiplying large numbers and then having to simplify a large fraction. The final answer comes out already in lowest terms.
Can you show an example of reducing before multiplying?
In (8/9) × (3/4): 8 and 4 share factor 4 (→2 and 1), 9 and 3 share factor 3 (→3 and 1). Result: (2/3) × (1/1) = 2/3.
Where is reducing before multiplying taught in Saxon Math Course 2?
This technique is covered in Saxon Math, Course 2, as part of Grade 7 fraction multiplication and simplification content.
Is reducing before multiplying the same as cross-canceling?
Yes, they are the same procedure. 'Cross-canceling' describes the diagonal pattern when you cancel a numerator from one fraction with a denominator from another.
What if no factors can be canceled before multiplying?
If the numerators and denominators share no common factors, simply multiply straight across, then simplify the result if needed.