Finding the unknown total
Finding the unknown total means working backward from a known part and its fraction of the whole to determine the complete quantity. If you know that 2/3 of a group equals 18, divide by the numerator to find one part (18 divided by 2 = 9), then multiply by the denominator to get the whole (9 times 3 = 27). This Grade 7 math skill from Saxon Math, Course 2 is the algebraic inverse of finding a fractional part of a number, and it appears in percent-of-total problems, scientific sampling, and survey data interpretation.
Key Concepts
Property When the total number is the unknown, set up the equation using a variable for the total: $\operatorname{Fraction} \times W N = \operatorname{Part}$.
Examples Two thirds of what number is 50? $\rightarrow \frac{2}{3} \times W N = 50 \rightarrow W N = 50 \times \frac{3}{2} = 75$. Sixty is 0.4 of what number? $\rightarrow 60 = 0.4 \times W N \rightarrow W N = \frac{60}{0.4} = 150$. Five fourths of what number is 25? $\rightarrow \frac{5}{4} \times W N = 25 \rightarrow W N = 25 \times \frac{4}{5} = 20$.
Explanation To find the unknown total, you must free it from the fraction! You can do this by multiplying both sides by the fraction's reciprocal, which cancels it out and reveals the answer. If you are working with a decimal instead of a fraction, simply divide both sides by it.
Common Questions
How do I find the unknown total when a fraction of it is known?
Divide the known amount by the fraction's numerator to find the value of one unit, then multiply by the denominator. If 2/3 of a number is 18, one-third is 18 divided by 2 = 9, and the whole is 9 times 3 = 27.
What equation can I use to find the unknown total?
Let W be the whole. Set up the equation: (fraction) times W = (known amount). Then solve for W by dividing both sides by the fraction. For example, (2/3) times W = 18 gives W = 18 divided by (2/3) = 27.
How is finding the unknown total different from finding a fractional part?
Finding a fractional part starts with the whole and finds a portion. Finding the unknown total starts with the portion and works back to the whole — the two operations are inverses of each other.
Why is finding the unknown total important?
This skill is used in real-world contexts like finding the original price before a discount, the full population from a sample count, or the total items when you know a fraction of them.
What are common mistakes when finding the unknown total?
Students often multiply by the numerator when they should divide, or skip the step of finding the unit value first. Using a clear equation helps avoid mixing up the steps.
When do students learn to find the unknown total?
Finding the unknown total is typically a Grade 7 skill. Saxon Math, Course 2 covers it in Chapter 6 as part of fraction and proportion problem-solving.
How does this skill connect to percent problems?
If 40% of a number is 20, you can find the total the same way: 20 divided by 0.4 = 50. Both fraction and percent versions of this problem use the same inverse reasoning.