Lesson 77: Finding Unstated Information in Fraction Problems

New Concept

Often fractional-parts statements contain more information than what is directly stated.

What’s next

Next, you will diagram these statements and answer questions to find all the hidden information.

A fractional-parts statement, such as 'Three fourths of the 28 students are boys,' indirectly provides information about the remaining part. The denominator indicates the total number of equal parts, while the numerator specifies the number of parts belonging to one group, allowing you to deduce the size and composition of the other group.

Three fifths of 30 students are girls. How many are boys? $\frac{3}{5} \times 30 = 18$ girls. Total students $30 - 18 = 12$ boys. In a bag of 48 marbles, $\frac{5}{8}$ are blue. How many are not blue? $\frac{5}{8} \times 48 = 30$ blue. Total marbles $48 - 30 = 18$ not blue. Of 40 little engines, $\frac{3}{8}$ could climb. How many could not? $\frac{3}{8} \times 40 = 15$ could climb. Total engines $40 - 15 = 25$ could not climb.

Think of a fraction problem as a detective clue! When it says 'three-fourths of students are boys,' it’s not just about the boys. The denominator (4) tells you the whole group is split into four equal teams, and the numerator (3) says three teams are boys. This secretly reveals there is one team of girls, letting you find their exact number!

The probability of an event and its complement (the event not happening) always sum to 1. If the probability of an event is $P(A)$, the probability of its complement is $P(\text{not } A) = 1 - P(A)$. For example, if $P(\text{red}) = \frac{2}{5}$, then $P(\text{not red}) = 1 - \frac{2}{5} = \frac{3}{5}$.

The probability of rain is $\frac{3}{10}$. What is the probability of no rain? $1 - \frac{3}{10} = \frac{7}{10}$. A spinner has a $\frac{1}{4}$ chance of landing on blue. What is the chance of not landing on blue? $1 - \frac{1}{4} = \frac{3}{4}$. If the probability of winning is $\frac{2}{7}$, the probability of not winning is $1 - \frac{2}{7} = \frac{5}{7}$.

Imagine you have a bag of mystery jellybeans. The chance of picking a gross flavor is one thing, and the chance of picking anything but a gross flavor is its 'complement,' or opposite. Since those are the only two outcomes, their probabilities must add up to a perfect 1 (or 100%). It’s a handy shortcut to find the odds of something not happening!