Bottle 1 contains water and cordial in the ratio 2 : 5. Bottle 2 has the same total volume and has water and cordial in the ratio 3 : 7. If the contents are combined, what is the ratio of water to cordial? Hint: It is not 5 : 12

Let's solve the problem step by step.

### Step 1: Define the ratios and volumes

- **Bottle 1**: Water to cordial ratio is 2:5. Let the volume of Bottle 1 be \( V_1 \).
- **Bottle 2**: Water to cordial ratio is 3:7. Let the volume of Bottle 2 be \( V_2 \).

Since the problem states that both bottles have the same total volume, we can say:

\[
V_1 = V_2 = V
\]

### Step 2: Express the amounts of water and cordial

For **Bottle 1**:
- Water amount = \( \frac{2}{7} \times V \)
- Cordial amount = \( \frac{5}{7} \times V \)

For **Bottle 2**:
- Water amount = \( \frac{3}{10} \times V \)
- Cordial amount = \( \frac{7}{10} \times V \)

### Step 3: Calculate the total amount of water and cordial when the bottles are combined

- **Total Water** = Water from Bottle 1 + Water from Bottle 2
  \[
  \text{Total Water} = \frac{2}{7}V + \frac{3}{10}V
  \]

- **Total Cordial** = Cordial from Bottle 1 + Cordial from Bottle 2
  \[
  \text{Total Cordial} = \frac{5}{7}V + \frac{7}{10}V
  \]

### Step 4: Find a common denominator and simplify

- For water:
  \[
  \text{Total Water} = \frac{2 \times 10}{70}V + \frac{3 \times 7}{70}V = \frac{20}{70}V + \frac{21}{70}V = \frac{41}{70}V
  \]

- For cordial:
  \[
  \text{Total Cordial} = \frac{5 \times 10}{70}V + \frac{7 \times 7}{70}V = \frac{50}{70}V + \frac{49}{70}V = \frac{99}{70}V
  \]

### Step 5: Calculate the ratio of water to cordial

The ratio of water to cordial is:
\[
\text{Ratio} = \frac{\frac{41}{70}V}{\frac{99}{70}V} = \frac{41}{99}
\]

So, the combined ratio of water to cordial is **41:99**.

Would you like any details or have any questions?

Here are 5 related questions for further exploration:
1. What would the ratio be if Bottle 2 had twice the volume of Bottle 1?
2. How does the ratio change if the water content in Bottle 1 increases?
3. If both bottles had a different ratio, how would you approach the problem?
4. How does the ratio of cordial to water affect the taste of the mixture?
5. How can you adjust the ratio if the mixture is too concentrated?

**Tip:** When combining mixtures with different ratios, always express the quantities in terms of a common denominator to simplify the calculations.

Explore how to find the ratio of water to cordial when combining two bottles with different water-cordial ratios. Step-by-step solution provided with calculations. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation