Equivalent Division Problems
Equivalent division problems with decimals are created by multiplying both the dividend and divisor by the same power of 10 to eliminate decimal divisors. For example, 1.36 / 0.4 becomes 13.6 / 4 when both are multiplied by 10, and 7 / 0.35 becomes 700 / 35 = 20 when both are multiplied by 100. This technique is covered in Chapter 5 of Saxon Math Course 2 for 7th grade math and is essential for dividing by decimals without a calculator, transforming tricky decimal division into straightforward whole-number problems.
Key Concepts
Property You can create an equivalent division problem by multiplying the dividend and divisor by the same number. It's the same principle as multiplying a fraction by a form of 1, like $\frac{10}{10}$.
Examples $\frac{1.36}{0.4}$ becomes $\frac{1.36 \times 10}{0.4 \times 10}$ = $\frac{13.6}{4}$ $\frac{7}{0.35}$ becomes $\frac{7.00 \times 100}{0.35\times100}$ = $\frac{700}{35}$ = 20.
Explanation This is the secret power behind our decimal trick! Multiplying the top and bottom of a division problem by 10 or 100 doesn't change the final answer. It just cleverly transforms the problem into a friendlier version that gets rid of the pesky decimal in the number you're dividing by.
Common Questions
How do you divide by a decimal number?
Multiply both the dividend and divisor by the same power of 10 to make the divisor a whole number. For 1.36 / 0.4, multiply both by 10 to get 13.6 / 4 = 3.4. This creates an equivalent problem that is much easier to solve.
Why do you multiply by 10 when dividing decimals?
Multiplying both numbers by 10 (or 100) moves the decimal point and turns the divisor into a whole number. Since you multiply both by the same number, the quotient stays the same. This is like multiplying a fraction by 10/10, which equals 1.
How do you decide whether to multiply by 10, 100, or 1000?
Count the decimal places in the divisor. If it has one decimal place (like 0.4), multiply by 10. Two decimal places (like 0.35) means multiply by 100. The goal is to make the divisor a whole number.
What is an example of equivalent division with decimals?
To solve 7 / 0.35, multiply both by 100: 700 / 35 = 20. The original problem and the new one have the same answer because you multiplied numerator and denominator by the same factor.
What is the connection between equivalent division and fractions?
A division problem is a fraction. Creating an equivalent division problem is like creating an equivalent fraction by multiplying numerator and denominator by the same value. For instance, 1.36/0.4 = 13.6/4, just as 2/3 = 4/6.
When do students learn to divide by decimals?
Dividing by decimals using equivalent division is taught in 7th grade math. Saxon Math Course 2 covers this in Chapter 5, building on the equivalent division concept introduced in Chapter 3 by applying it specifically to decimal divisors.