Fraction of a quantity
Two thirds of a 120 mile trip, or three fourths of a mural you can only see part of. How do you find a fraction of an amount, and how do you work back to the whole?
Where you are 3 inside
What this area is about
Every question here ties three numbers together: \text{part} = \text{fraction} \times \text{whole}. The word “of” means times. Given the fraction and the whole, you find the part; given the part and the fraction, you find the whole. A tape diagram, a bar cut into equal pieces, shows both directions.
How these fit together
Finding the part. A fraction of a whole number splits the amount into equal groups and takes some of them: \displaystyle \frac{a}{b} of N is N \div b, taken a times, for whole numbers a, N and b \neq 0.
Two cuts in a row. A fraction of a fraction of an amount applies a second fraction to the part the first one took, which multiplies the two fractions, and also handles a second cut of the rest.
Working backward. Finding the whole starts from a known part and its fraction: divide by the numerator to get one piece, then multiply by the denominator. This needs a fraction whose numerator is not 0.
A path through it
A fraction of a whole number comes first. The other two build on it, in either order.
This area feeds percents and ratios: finding a percent of a quantity and tape diagrams for sharing in a ratio both reuse the same part, fraction and whole.