Solving Ratio Problems: Finding Ratios of Expressions

This problem involves finding the ratio of two algebraic expressions when the ratios of the variables are given. We are given the ratio \(a : b : c = 5 : 6 : 4\) and asked to find the value of the ratio \((a + b + c) : (3a + b - c)\).

When variables are in a given ratio, like \(a : b : c = 5 : 6 : 4\), it means that \(a\), \(b\), and \(c\) are proportional to 5, 6, and 4 respectively. We can represent the variables using a common constant multiplier, say \(k\). So, we can write:

• \(a = 5k\)
• \(b = 6k\)
• \(c = 4k\)

where \(k\) is a non-zero constant.

Now, we need to find the values of the two expressions involved in the ratio \((a + b + c) : (3a + b - c)\) by substituting these values of \(a\), \(b\), and \(c\).

Step 1: Calculate the value of the first expression \((a + b + c)\).

Substitute \(a = 5k\), \(b = 6k\), and \(c = 4k\) into the expression:

\(a + b + c = (5k) + (6k) + (4k)\)

Combine the terms:

\(a + b + c = (5 + 6 + 4)k = 15k\)

So, the value of the first expression is \(15k\).

Step 2: Calculate the value of the second expression \((3a + b - c)\).

Substitute \(a = 5k\), \(b = 6k\), and \(c = 4k\) into the second expression:

\(3a + b - c = 3(5k) + (6k) - (4k)\)

Perform the multiplication and then combine the terms:

\(3(5k) + (6k) - (4k) = 15k + 6k - 4k\)

\(15k + 6k - 4k = (15 + 6 - 4)k\)

\((15 + 6 - 4)k = (21 - 4)k = 17k\)

So, the value of the second expression is \(17k\).

Step 3: Find the ratio \((a + b + c) : (3a + b - c)\).

Now we have the values of the two expressions in terms of \(k\):

First expression value: \(15k\)

Second expression value: \(17k\)

The ratio is the first value divided by the second value:

\((a + b + c) : (3a + b - c) = \frac{15k}{17k}\)

Since \(k\) is a non-zero constant, we can cancel \(k\) from the numerator and the denominator:

\(\frac{15k}{17k} = \frac{15}{17}\)

So, the ratio \((a + b + c) : (3a + b - c)\) is \(15 : 17\).

Therefore, the value of \((a + b + c) : (3a + b - c)\) is \(15 : 17\).

Let's quickly check the options provided:

• 18 : 23
• 25 : 23
• 18 : 19
• 15 : 17

Our calculated ratio \(15 : 17\) matches one of the options.

Revision Table: Key Concepts for Ratio Problems

Understanding the basics of ratios is crucial for solving these types of problems. Here's a quick recap:

Additional Information: Working with Ratios and Expressions

When dealing with ratios involving multiple variables and expressions, the method of using a common multiple (\(k\) in our case) is very effective. This technique allows you to convert the proportional relationship into algebraic expressions that can be manipulated.

• Why use a constant \(k\)? The ratio \(a : b : c = 5 : 6 : 4\) means that \(a/5 = b/6 = c/4\). Let this common value be \(k\). Then \(a/5 = k \implies a = 5k\), \(b/6 = k \implies b = 6k\), and \(c/4 = k \implies c = 4k\). This confirms our representation.
• Substituting into Expressions: Once you have the variables in terms of \(k\), substitute these into the required expressions. This turns the expressions of variables into expressions of \(k\).
• Forming the Ratio: The ratio of two expressions is simply the value of the first expression divided by the value of the second expression.
• Canceling the Constant: In the final ratio, the constant \(k\) will usually cancel out, resulting in a simple numerical ratio. This is because the ratio depends on the proportions of the variables, not their absolute values. If \(k\) didn't cancel, it might indicate an error, or that the ratio depends on the magnitude (though typically in such problems, it cancels).

This method is widely applicable to ratio problems involving sums, differences, or other linear combinations of the variables.