Grade Distribution and Population

The question asks for the least possible population of a class given the distribution of grades as fractions. The population of the class must be a whole number (integer), and the number of students receiving each grade must also be a whole number.

The table shows the fraction of the population for each grade:

Fractions and Population Size

Let the total population of the class be $P$. The number of students receiving each grade is calculated by multiplying the total population by the fraction for that grade:

• Number of students with Grade A = $0.1 \times P$
• Number of students with Grade B = $0.4 \times P$
• Number of students with Grade C = $0.3 \times P$
• Number of students with Grade D = $0.2 \times P$

For the number of students in each grade to be an integer, $P$ must be a number such that when multiplied by each fraction ($0.1, 0.4, 0.3, 0.2$), the result is a whole number. This means $P$ must be divisible by the denominators of these fractions when written in their simplest form.

Converting Decimals to Fractions

Let's convert the decimal fractions to common fractions:

• $0.1 = \frac{1}{10}$
• $0.4 = \frac{4}{10} = \frac{2}{5}$
• $0.3 = \frac{3}{10}$
• $0.2 = \frac{2}{10} = \frac{1}{5}$

The denominators of these fractions are 10, 5, 10, and 5. To find the least possible population $P$, $P$ must be the smallest number that is a multiple of all these denominators. This is the Least Common Multiple (LCM) of the denominators 10 and 5.

Finding the Least Common Multiple (LCM)

We need to find the LCM of 10 and 5.

• Multiples of 10: 10, 20, 30, 40, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...

The least common multiple of 10 and 5 is 10.

Therefore, the least possible population of the class is 10.

Verifying the Population

Let's check if a population of 10 results in an integer number of students for each grade:

• Grade A: $0.1 \times 10 = 1$ student
• Grade B: $0.4 \times 10 = 4$ students
• Grade C: $0.3 \times 10 = 3$ students
• Grade D: $0.2 \times 10 = 2$ students

The number of students for each grade (1, 4, 3, 2) are all integers. The total number of students is $1 + 4 + 3 + 2 = 10$, which matches the population.

Any smaller integer population would result in a non-integer number of students for at least one grade. For example, if the population was 5:

• Grade A: $0.1 \times 5 = 0.5$ (not an integer)

Thus, the least possible population is 10.