The average of 4, 6, 8, 12 and x is 7 and the average of x, 9, 13, 15 and y is 9. What is the value of 2x - 3y?
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Solving Average Problems to Find Unknowns
This problem requires us to use the concept of average to find the values of two unknown variables, 'x' and 'y', and then evaluate a given expression involving these variables.
The average of a set of numbers is calculated by summing all the numbers in the set and then dividing by the total count of numbers in the set. The formula for average is:
\text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}
Finding the Value of x
We are given that the average of the numbers 4, 6, 8, 12, and x is 7. There are 5 numbers in this set. Using the average formula, we can set up an equation:
\frac{4 + 6 + 8 + 12 + x}{5} = 7
First, sum the known numbers:
4 + 6 + 8 + 12 = 30
Substitute this sum back into the equation:
\frac{30 + x}{5} = 7
To solve for x, multiply both sides of the equation by 5:
30 + x = 7 \times 5
30 + x = 35
Now, subtract 30 from both sides to isolate x:
x = 35 - 30
x = 5
So, the value of x is 5.
Finding the Value of y
We are also given that the average of the numbers x, 9, 13, 15, and y is 9. There are 5 numbers in this second set. We already found that x = 5. Using the average formula for the second set:
\frac{x + 9 + 13 + 15 + y}{5} = 9
Substitute the value of x (which is 5) into this equation:
\frac{5 + 9 + 13 + 15 + y}{5} = 9
Sum the known numbers in the numerator:
5 + 9 + 13 + 15 = 42
Substitute this sum back into the equation:
\frac{42 + y}{5} = 9
Multiply both sides of the equation by 5:
42 + y = 9 \times 5
42 + y = 45
Subtract 42 from both sides to isolate y:
y = 45 - 42
y = 3
So, the value of y is 3.
Evaluating the Expression 2x - 3y
Now that we have the values of x and y, we can find the value of the expression 2x - 3y. We found x = 5 and y = 3.
Substitute these values into the expression:
2x - 3y = 2(5) - 3(3)
Perform the multiplication:
2(5) = 10
3(3) = 9
Substitute these results back into the expression:
2x - 3y = 10 - 9
Perform the subtraction:
10 - 9 = 1
Therefore, the value of 2x - 3y is 1.
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Find sum of first set (excluding x) | $4 + 6 + 8 + 12$ | 30 |
| 2 | Set up equation for first average | $\frac{30 + x}{5} = 7$ | |
| 3 | Solve for x | $30 + x = 35 \Rightarrow x = 35 - 30$ | $x = 5$ |
| 4 | Find sum of second set (excluding y, using x) | $x + 9 + 13 + 15 = 5 + 9 + 13 + 15$ | 42 |
| 5 | Set up equation for second average | $\frac{42 + y}{5} = 9$ | |
| 6 | Solve for y | $42 + y = 45 \Rightarrow y = 45 - 42$ | $y = 3$ |
| 7 | Evaluate $2x - 3y$ | $2(5) - 3(3) = 10 - 9$ | 1 |
Revision Table: Key Concepts
| Concept | Definition/Application |
|---|---|
| Average (Mean) | Sum of a set of values divided by the number of values. Crucial for setting up the initial equations. |
| Algebraic Equations | Mathematical sentences stating that two expressions are equal. Used to represent the given average conditions. |
| Solving Linear Equations | Techniques like addition, subtraction, multiplication, and division applied to both sides of an equation to isolate the unknown variable (x and y). |
| Substitution | Replacing a variable with its known value in an expression or another equation. Used to find y after finding x, and to evaluate $2x - 3y$. |
Additional Information: Understanding Average
The average, or arithmetic mean, is a measure of central tendency. It gives us a single value that represents the typical value of a set of numbers. It's widely used in statistics, data analysis, and everyday calculations.
When calculating the average, every number in the set contributes to the sum. In our problem, the variables x and y are just like any other number in their respective sets, influencing the total sum and thus the average.
Solving equations involving averages often requires basic algebraic skills, such as combining like terms and performing inverse operations to isolate the variable. In this case, we used multiplication and subtraction to solve for x and y.
Understanding how to work with averages and solve simple algebraic equations is fundamental for many types of mathematical problems.
Important Questions from Average
The average height of 20 students of class 8 is 152 cm and the average height of 15 students of class 9 is 168 cm. What is the average height (to the nearest cm) of the students of both classes?
The average weight of 20 girls in a school was 52 kg. Two new students of weight 54 kg and 50 kg were admitted. The ratio of this new average to the old one is:
If the average of two numbers is 13 and the square root of their product is 12, then the difference between the numbers is:
If the average of 5 consecutive odd integers in increasing order is 11 , then the average of the last 3 of them is:
There are two sections A and B of a class, consisting of 38 and 42 students respectively. if the average weight of the students of section A is 55 Kg and that of section B is 32 Kg. find the average weight of all the students in the class.
