 Hema bought some chocolates and gave half of them to Shireen. Shireen bought some sweets and gave half of them to Hema. Hema ate 10 sweets and Shireen ate 4 chocolates. After that, the number of sweets and chocolates Hema had were in the ratio 6:5 and the number of sweets and chocolates Shireen had were in the ratio of 11:3. How many chocolates did Hema buy?

Let's assume the following:

- Hema initially bought \( x \) chocolates.
- Shireen initially bought \( y \) sweets.

### Step 1: Analyzing Hema's situation

- Hema gives half of her chocolates to Shireen, so she is left with \( \frac{x}{2} \) chocolates.
- Shireen gives half of her sweets to Hema, so Hema receives \( \frac{y}{2} \) sweets.
- Hema eats 10 sweets, so the number of sweets Hema has after eating is \( \frac{y}{2} - 10 \).
- The number of chocolates Hema has doesn't change, so she still has \( \frac{x}{2} \) chocolates.

According to the problem, after all this, the ratio of sweets to chocolates that Hema has is 6:5. Therefore, we can write the equation:

\[
\frac{\frac{y}{2} - 10}{\frac{x}{2}} = \frac{6}{5}
\]

Multiplying both sides by \( \frac{x}{2} \) to eliminate the denominator:

\[
\frac{y}{2} - 10 = \frac{6}{5} \times \frac{x}{2}
\]

Simplifying:

\[
\frac{y}{2} - 10 = \frac{3x}{5}
\]

### Step 2: Analyzing Shireen's situation

- Shireen receives half of Hema's chocolates, so she gets \( \frac{x}{2} \) chocolates.
- Shireen eats 4 chocolates, so she is left with \( \frac{x}{2} - 4 \) chocolates.
- Shireen gives half of her sweets to Hema, so she is left with \( \frac{y}{2} \) sweets.

The problem states that the ratio of sweets to chocolates that Shireen has is 11:3. Therefore, we can write the equation:

\[
\frac{\frac{y}{2}}{\frac{x}{2} - 4} = \frac{11}{3}
\]

Multiplying both sides by \( \frac{x}{2} - 4 \):

\[
\frac{y}{2} = \frac{11}{3} \times \left( \frac{x}{2} - 4 \right)
\]

Simplifying:

\[
\frac{y}{2} = \frac{11}{6} \times (x - 8)
\]

### Step 3: Solving the system of equations

We now have two equations:

1. \( \frac{y}{2} - 10 = \frac{3x}{5} \)
2. \( \frac{y}{2} = \frac{11}{6} \times (x - 8) \)

Let's solve these equations step by step.

#### From equation (1):
\[
\frac{y}{2} = \frac{3x}{5} + 10
\]

#### Substituting this into equation (2):

\[
\frac{3x}{5} + 10 = \frac{11}{6} \times (x - 8)
\]

Multiplying both sides by 30 to eliminate the denominators:

\[
6 \times (3x + 50) = 55 \times (x - 8)
\]

Expanding both sides:

\[
18x + 300 = 55x - 440
\]

Bringing like terms together:

\[
300 + 440 = 55x - 18x
\]

\[
740 = 37x
\]

Solving for \( x \):

\[
x = \frac{740}{37} = 20
\]

### Step 4: Finding the number of sweets \( y \)

Substitute \( x = 20 \) into the equation \( \frac{y}{2} = \frac{3x}{5} + 10 \):

\[
\frac{y}{2} = \frac{3 \times 20}{5} + 10 = 12 + 10 = 22
\]

So,

\[
y = 44
\]

### Conclusion:

Hema initially bought **20 chocolates**.

Would you like more details or have any questions?

Here are 5 relative questions that extend this problem:
1. What would happen if Hema bought 30 chocolates instead?
2. How would the solution change if Shireen ate 6 chocolates instead of 4?
3. What if the ratio of sweets to chocolates Hema had was 7:5 instead of 6:5?
4. How does changing the number of sweets Shireen buys affect the final ratios?
5. What if Shireen gave 1/3 of her sweets to Hema instead of half?

**Tip**: When dealing with ratios, always express both quantities in the same form (like fractions) to simplify solving equations.

Hema bought some chocolates and gave half of them to Shireen. Shireen bought some sweets and gave half of them to Hema. Hema ate 10 sweets and Shireen ate 4 chocolates. After that, the number of sweets and chocolates Hema had were in the ratio 6:5 and the number of sweets and chocolates Shireen had were in the ratio of 11:3. How many chocolates did Hema buy?

Explore how to solve a ratio and proportion problem involving Hema and Shireen's chocolates and sweets. This problem uses algebraic methods to find out how many chocolates Hema initially bought. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation