Question

If the average (arithmetic mean) of \(a\) and \(b\) is 45 and the average of \(b\) and \(c\) is \(70,\) what is the value of \(c-a ?\)

Short answer

The value of \(c - a\) is 50.

Step-by-step solution

Understanding the Average

The average (arithmetic mean) of two numbers is calculated by summing the numbers and then dividing by 2. The given average for \(a\) and \(b\) is 45. Therefore, we have the equation: \[ \frac{a + b}{2} = 45 \]

Solve for a + b

To clear the fraction, multiply both sides by 2: \[ a + b = 90 \]

Understanding the Second Average

Similarly, the average of \(b\) and \(c\) is given as 70. Therefore, we have the equation: \[ \frac{b + c}{2} = 70 \]

Solve for b + c

To clear the fraction, multiply both sides by 2: \[ b + c = 140 \]

Subtract the Equations

Subtract the first equation from the second equation to eliminate \(b\): \[ (b + c) - (a + b) = 140 - 90 \]

Simplify

Simplify the resulting equation: \[ c - a = 50 \]

Key concepts

Arithmetic Mean

The arithmetic mean, often just called the average, is a key concept in mathematics. It represents the central value of a set of numbers.
To find the arithmetic mean of two numbers, you sum the numbers and then divide by the count of the numbers.
For example, if you want to find the average of two numbers, like in our problem where the average of \(a\) and \(b\) is given as 45, you use the formula:
\[ \text{Average} = \frac{a + b}{2} = 45 \.\]
Rewriting this equation to solve for \(a + b\), you multiply both sides by 2 to get:
\[ a + b = 90 \]
Keep this basic process in mind whenever you need to calculate an arithmetic mean. It will be very useful for solving various types of problems.

Linear Equations

Linear equations are fundamental in problem-solving and algebra. They are equations that form a straight line when graphed, and they can always be written in the form \(ax + by = c\).
In our problem, we used two linear equations stemming from the given averages:
For \(a\) and \(b\)
\[ a + b = 90 \]
For \(b\) and \(c\)
\[ b + c = 140 \]
To find the value of \(c - a\), we can use subtraction to eliminate the common variable \(b\):
Subtract the equation for \(a + b\) from \(b + c\):
\[ (b + c) - (a + b) = 140 - 90 \]
Simplify to get:
\[ c - a = 50 \]
These steps highlight how linear equations can be combined to solve for unknown variables. Mastering this technique is crucial for tackling more complex problems.

Problem-Solving Steps

Problem solving in mathematics often follows a series of logical steps. Here is a structured approach we used in solving our given problem:

• Step 1: Understand the problem. Identify what is given and what needs to be found. In our case, we were given the averages of \(a\) and \(b\) and \(b\) and \(c\), and we needed the difference \(c - a\).
• Step 2: Translate the words into algebraic equations. Use the definition of arithmetic mean to create equations.
• Step 3: Solve the equations. First, solve for the sums like \(a + b\) and \(b + c\). Then manipulate the equations to isolate the unknowns.
• Step 4: Simplify your results. Combine equations if necessary. In our problem, subtracting them simplified the solution.
• Step 5: Verify your solution. Check that the solution makes sense in the context of the problem.

Following these structured steps can help tackle a wide range of math problems. Consistently practice this approach to improve your problem-solving skills.