Solving Percentage and Equation Problems

This problem involves understanding percentages and solving a linear equation to find the values of three variables, A, B, and C, and then evaluating a given expression.

Let's break down the information given:

• A is 120% of B
• B is 65% of C
• The sum of A, B, and C is 121.5

We need to find the value of the expression $\text{C} - 2\text{B} + \text{A}$.

Step 1: Expressing Variables in Terms of C

First, let's convert the percentage relationships into mathematical equations. Remember that a percentage can be written as a decimal by dividing by 100.

• B is 65% of C means $\text{B} = \frac{65}{100} \times \text{C} = 0.65\text{C}$.
• A is 120% of B means $\text{A} = \frac{120}{100} \times \text{B} = 1.20\text{B}$.

Now, we can express A in terms of C by substituting the value of B from the first equation into the second equation:

$\text{A} = 1.20 \times \text{B}$

$\text{A} = 1.20 \times (0.65\text{C})$

$\text{A} = (1.20 \times 0.65)\text{C}$

Let's calculate $1.20 \times 0.65$:

So, $\text{A} = 0.78\text{C}$.

Now we have both A and B expressed in terms of C:

• $\text{A} = 0.78\text{C}$
• $\text{B} = 0.65\text{C}$

Step 2: Using the Sum to Find C

We are given that the sum of A, B, and C is 121.5:

$\text{A} + \text{B} + \text{C} = 121.5$

Substitute the expressions for A and B in terms of C into this equation:

$(0.78\text{C}) + (0.65\text{C}) + \text{C} = 121.5$}

Combine the terms with C:

$(0.78 + 0.65 + 1)\text{C} = 121.5$

$(1.43 + 1)\text{C} = 121.5$

$2.43\text{C} = 121.5$

Now, solve for C:

$\text{C} = \frac{121.5}{2.43}$

To make the division easier, we can multiply the numerator and denominator by 100 to remove the decimal points:

$\text{C} = \frac{121.5 \times 100}{2.43 \times 100} = \frac{12150}{243}$

Let's perform the division:

So, $\text{C} = 50$.

Step 3: Finding the Values of A and B

Now that we know the value of C, we can find the values of B and A using the relationships we found earlier:

• $\text{B} = 0.65\text{C} = 0.65 \times 50 = 32.5$
• $\text{A} = 0.78\text{C} = 0.78 \times 50 = 39$

Let's verify the sum: $\text{A} + \text{B} + \text{C} = 39 + 32.5 + 50 = 121.5$. This matches the given information, so our values for A, B, and C are correct.

Step 4: Evaluating the Expression C - 2B + A

Finally, we need to calculate the value of the expression $\text{C} - 2\text{B} + \text{A}$ using the values we found for A, B, and C:

$\text{C} - 2\text{B} + \text{A} = 50 - 2(32.5) + 39$

First, calculate $2\text{B}$:

$2\text{B} = 2 \times 32.5 = 65$

Now substitute this back into the expression:

$50 - 65 + 39$

Perform the subtraction and addition:

$50 - 65 = -15$

$-15 + 39 = 24$

The value of $\text{C} - 2\text{B} + \text{A}$ is 24.

This value matches one of the given options.

Revision Table: Key Steps for Percentage Problems

Additional Information on Percentages and Variables

Percentages are a way of expressing a fraction out of 100. For example, 65% means 65 out of 100, or $\frac{65}{100}$. When you see "percent of" in a word problem, it usually indicates multiplication.

Problems involving percentages and multiple variables often require setting up a system of equations. In this case, we had implicit equations from the percentage relationships and an explicit equation from the sum.

By expressing all variables in terms of one common variable (like C in this case), we simplified the system into a single equation with one unknown, which is easier to solve. This is a common strategy in algebra problems.

The expression we needed to evaluate, $\text{C} - 2\text{B} + \text{A}$, is a linear combination of the variables. Once the values of the individual variables are known, evaluating such an expression is a straightforward substitution and arithmetic calculation.