Equivalent division problems
Equivalent division problems are created by multiplying or dividing both the dividend and divisor by the same number, producing a new problem with the same quotient that may be easier to solve mentally. For example, 4.00 / 16 can be simplified by halving both: 2.00 / 8, then 1.00 / 4 = 0.25. Similarly, 220 / 5 becomes 440 / 10 = 44 when both are doubled. This technique is taught in Chapter 3 of Saxon Math Course 2 for 7th grade math and builds mental math fluency and number sense for division.
Key Concepts
Property We multiply (or divide) the dividend and divisor by the same number to form a new division problem that is easier to calculate mentally. The new division problem will produce the same quotient. $$ \frac{a \div c}{b \div c} = \frac{a}{b} \quad \text{and} \quad \frac{a \cdot c}{b \cdot c} = \frac{a}{b} $$.
Examples To solve 4.00 dollars $\div$ 16, divide both by 2 to get 2.00 dollars $\div$ 8. Divide by 2 again to get 1.00 dollar $\div$ 4 = 0.25 dollars. Instead of $220 \div 5$, double both numbers to get a simpler problem: $440 \div 10 = 44$. The answer is the same! To solve $24,000 \div 400$, divide both numbers by 100 to simplify it. This creates the problem $240 \div 4 = 60$.
Explanation Stuck on a tricky division problem? Just change the numbers! You can multiply or divide both parts of the problem by the same amount, and the final answer stays the same. It is like zooming in or out on a map—the destination does not change! This trick can turn a tough calculation into one you can solve in your head.
Common Questions
What are equivalent division problems?
Equivalent division problems are different division expressions that produce the same quotient. You create them by multiplying or dividing both the dividend and divisor by the same nonzero number, like turning 220 / 5 into 440 / 10 = 44.
How do you create an equivalent division problem?
Multiply or divide both the dividend and divisor by the same number. This is equivalent to multiplying the fraction by a form of 1. For example, multiply both parts of 4.00 / 16 by 1/2 to get 2.00 / 8, which is easier to compute.
Why are equivalent division problems useful?
They let you replace a difficult division with an easier one. Dividing by 5 is hard, but doubling both numbers lets you divide by 10 instead, which you can do by just moving the decimal point. This builds mental math speed.
How is this related to fractions?
A division problem is the same as a fraction. Creating an equivalent division problem is like simplifying or scaling a fraction. For example, 4/16 = 2/8 = 1/4 = 0.25. The value never changes because you multiply by 1.
What are common strategies for equivalent division?
Divide both by 2 to simplify even numbers, multiply both by 2 when dividing by 5 (to divide by 10 instead), or multiply both by 10 to eliminate decimals in the divisor. Choose whichever makes the calculation easiest.
When do students learn equivalent division problems?
This skill is taught in 7th grade math. Saxon Math Course 2 introduces it in Chapter 3 and revisits it in Chapter 5, helping students develop mental math strategies for division.