If a : b : c = 3 : 4 : 7, then the ratio (a + b + c) : c is equal to

Question:

If a : b : c = 3 : 4 : 7, then the ratio (a + b + c) : c is equal to

A. 2 : 1
B. 14 : 3
C. 7 : 2
D. 1 : 2

Core Concept

This question is based on the concept of Ratios and Proportions. A ratio is a way of comparing two or more quantities in relation to each other. The basic idea is that ratios represent proportional relationships, which are expressed as fractions or as simple numbers.

In this case, the given ratio a : b : c = 3 : 4 : 7 means that for every 3 units of a, there are 4 units of b, and 7 units of c. The problem asks to find the ratio of the sum of a + b + c to c.

Step-by-Step Solution

The problem provides the ratio of three quantities:

$$a : b : c = 3 : 4 : 7$$

We are asked to find the ratio of $(a + b + c) : c$.

Step 1: Express the quantities in terms of a common variable

Let us assume a common multiple $k$ for the given ratio. So, we can write the values of a, b, and c as:

$$a = 3k, \quad b = 4k, \quad c = 7k$$

Step 2: Find the sum of a, b, and c

Now, we find the sum of a, b, and c:

$$a + b + c = 3k + 4k + 7k = 14k$$

Step 3: Set up the ratio $(a + b + c) : c$

The ratio we need to find is $(a + b + c) : c$. Substituting the values:

$$\text{Ratio} = \frac{a + b + c}{c} = \frac{14k}{7k}$$

Step 4: Simplify the ratio

Simplifying the above expression:

$$\text{Ratio} = \frac{14k}{7k} = 2$$

Thus, the ratio of $(a + b + c) : c$ is 2 : 1.

Correct Answer and Option

The correct answer is:
A. 2 : 1

Answer Reasoning

The ratio $(a + b + c) : c$ can be simplified as follows:

1. We expressed the quantities a, b, and c in terms of a common variable $k$.
2. The sum of a, b, and c was calculated as $14k$.
3. Dividing the sum by c, which is $7k$, we obtained the simplified ratio as 2 : 1.

Explanation of Incorrect Options

• B. 14 : 3: This is incorrect. This ratio does not reflect the relationship between the sum of a + b + c and c. The sum is 14k, but c is 7k, leading to a ratio of 2 : 1, not 14 : 3.
• C. 7 : 2: This is incorrect. It seems like the answer might have mistakenly swapped the terms or misunderstood the operations.
• D. 1 : 2: This is incorrect. The sum of a, b, and c is greater than c, so the ratio of the sum to c will be greater than 1, not 1 : 2.

Common Mistakes to Avoid

1. Misinterpreting the ratio: Some students may confuse the ratio a : b : c as direct values for calculation without introducing a common multiple $k$.
2. Incorrect simplification: It’s easy to make an error when simplifying the fraction. Ensure that both the numerator and denominator contain the same factor $k$ and that it cancels out correctly.

Simplest Way to Solve (Shortcut or Insight)

One shortcut to solve this type of problem is to:

• Write each of a, b, and c as multiples of a common variable $k$.
• Immediately sum them up and divide by c to get the ratio.
• If you’re familiar with ratios, this can be done mentally by focusing on the proportional relationships and simplifying directly.

For this case, the shortcut is:

$$\frac{a + b + c}{c} = \frac{14k}{7k} = 2 : 1$$

Visual Aid Suggestion

Here’s a simple visual to represent the ratio:

| Quantity   | a  | b  | c  |
|------------|----|----|----|
| Ratio      | 3k | 4k | 7k |
| Sum (a+b+c)| 14k|
| c          | 7k |

This table visually simplifies how the quantities are proportional to each other and helps clarify how we arrive at the ratio of $14k : 7k$.

Application

Understanding ratios and proportions is essential in real-life scenarios such as resource allocation, budgeting, and scaling. For example, in business, if three products a, b, and c have production ratios of 3 : 4 : 7, you can use the same method to calculate how the total amount of resources required for production compares to the amount needed for one of the products. This concept also applies in situations like investing, where different amounts of money are invested in various stocks, and you need to compare the total investment with a particular stock’s share.

Practice Questions

1. If x : y : z = 2 : 5 : 8, then the ratio (x + y + z) : z is equal to:
   • A. 10 : 1
   • B. 5 : 1
   • C. 3 : 1
   • D. 2 : 1
     (Answer: A)
2. If p : q : r = 4 : 7 : 9, then the ratio (p + q + r) : r is equal to:
   • A. 4 : 3
   • B. 3 : 2
   • C. 5 : 2
   • D. 7 : 3
     (Answer: C)
3. If a : b : c = 1 : 2 : 3, then the ratio (a + b + c) : c is equal to:
   • A. 3 : 1
   • B. 2 : 1
   • C. 5 : 2
   • D. 1 : 2
     (Answer: A)
4. If m : n : o = 5 : 6 : 7, then the ratio (m + n + o) : o is equal to:
   • A. 18 : 7
   • B. 11 : 7
   • C. 6 : 7
   • D. 5 : 7
     (Answer: B)
5. If a : b : c = 4 : 9 : 11, then the ratio (a + b + c) : b is equal to:
   • A. 7 : 5
   • B. 10 : 3
   • C. 6 : 5
   • D. 3 : 5
     (Answer: C)