Deducing the Number of Boys and Girls from a Given Ratio
Understanding ratios and proportions is a fundamental skill that can be applied in various real-life scenarios, such as determining the number of students of different genders in a class. This article will walk you through methods to find the number of boys and girls given their ratio and the total number of students. We will also discuss a common misunderstanding and provide a step-by-step solution.
Ratios and Proportions
A ratio is a comparison between two quantities, often expressed in the form of 'a to b' or 'a:b.' In this context, we are given a ratio of boys to girls in a class as 3:5, meaning for every 3 boys, there are 5 girls. Let's break down this concept with a few examples from the provided content.
Example 1 - Using Basic Division
The first approach is to break down the total number of students into parts based on the ratio. Given that there are 40 students in the class and the ratio is 3:5, we can deduce that one part of the ratio is represented by 5 students (5 parts of students). Therefore, three parts (boys) are:
3 parts times; 5 students/part 15 boys
And five parts (girls) are:
5 parts times; 5 students/part 25 girls
Adding these together confirms the total number of students:
15 boys 25 girls 40 students
Thus, there are 25 girls and 15 boys in the class.
Example 2 - Using Algebraic Substitution
A more formal approach involves setting up an equation using algebra. Let ( x ) be the number of boys and ( y ) be the number of girls. We know:
x y 40 (total students)
And the ratio is given as 3:5, which can be expressed as:
3x 5y
From the ratio equation, we can solve for ( x ) in terms of ( y ):
x frac{5}{3}y
Substituting ( x frac{5}{3}y ) into the total number equation:
frac{5}{3}y y 40
frac{5y 3y}{3} 40
frac{8y}{3} 40
y 40 times frac{3}{8}
y 15 (number of girls)
By substitution, we find:
x 40 - 15 25 (number of boys)
Therefore, the balance of the class is 25 boys and 15 girls.
Common Misunderstanding and Correction
The statement, "I don’t ‘do’ sums like everybody else," points to a common misunderstanding where students might overlook the fundamental steps of solving the problem. Solving the problem correctly involves breaking it down step-by-step, using ratio and proportion principles, and ensuring algebraic substitution is applied accurately.
Example 3 - Larger Class Size
Let's apply a similar method to the larger class size provided. Given 300 students in total, where (frac{3}{5}) are girls, we begin:
Number of girls frac{3}{5} times 300
180 girls
Subtracting the number of girls from the total number of students gives us the number of boys:
Number of boys 300 - 180 120
Therefore, in this class, there are 120 boys.
Conclusion
Understanding ratios and proportions is crucial in solving problems related to division and allocation. By breaking down the problem and applying basic arithmetic or algebraic substitution, the solution can be found accurately. Whether it's a small or larger class, the principles remain the same.