4 children shared some cards. Andy took ½ of the cards from the box first.
The ratio of the number of cards Sean took from the box to the number of remaining cards after Andy took from the box was | : 2. The rest of cards v shared by John and Frank. The ratio of the number of cards John took to th number of the rest of cards was 2: 3.
a The ratio of the number of cards Frank took to the number of cards Sea
iS
• (Express the ratio in its simplest form.)

Let's break down the problem step by step.

### Step 1: Let the total number of cards be \( x \).

- **Andy took** \( \frac{1}{2}x \) cards, leaving \( \frac{1}{2}x \) cards in the box.

### Step 2: Let the number of cards Sean took be \( y \).
  
- We're told that the ratio of the number of cards Sean took to the number of remaining cards after Andy took from the box was \( 1 : 2 \).
  
This means:
\[
\frac{y}{\text{Remaining Cards After Andy}} = \frac{1}{2}
\]

The remaining cards after Andy took are \( \frac{1}{2}x \), so:
\[
\frac{y}{\frac{1}{2}x} = \frac{1}{2}
\]
\[
y = \frac{1}{2} \times \frac{1}{2}x = \frac{1}{4}x
\]

So Sean took \( y = \frac{1}{4}x \) cards.

### Step 3: Now, the remaining cards after Sean took \( y \) cards will be:
\[
\frac{1}{2}x - \frac{1}{4}x = \frac{2}{4}x - \frac{1}{4}x = \frac{1}{4}x
\]

### Step 4: Let the number of cards John took be \( z \).

- We're told that the ratio of the number of cards John took to the number of the rest of the cards was \( 2 : 3 \).

This means:
\[
\frac{z}{\text{Remaining Cards}} = \frac{2}{3}
\]
The remaining cards after Sean took \( \frac{1}{4}x \) were \( \frac{1}{4}x \), so:
\[
\frac{z}{\frac{1}{4}x} = \frac{2}{3}
\]
\[
z = \frac{2}{3} \times \frac{1}{4}x = \frac{2}{12}x = \frac{1}{6}x
\]

### Step 5: The remaining cards after John took \( z \) cards will be:
\[
\frac{1}{4}x - \frac{1}{6}x = \frac{3}{12}x - \frac{2}{12}x = \frac{1}{12}x
\]
These remaining cards were taken by Frank.

### Step 6: The ratio of the number of cards Frank took to the number of cards Sean took is:
\[
\text{Ratio} = \frac{\frac{1}{12}x}{\frac{1}{4}x} = \frac{1}{12} \div \frac{1}{4} = \frac{1}{12} \times \frac{4}{1} = \frac{4}{12} = \frac{1}{3}
\]

### Final Answer:
The ratio of the number of cards Frank took to the number of cards Sean took is **1:3**.

Would you like further details or have any questions?

Here are some related questions:
1. How would the ratio change if Andy had taken 1/3 of the cards instead of 1/2?
2. What if the ratio of cards Sean took was 2:3 instead of 1:2?
3. How many cards would each child have if there were 48 cards in total?
4. What if John took 3/4 of the remaining cards instead of 2/3?
5. How would the distribution change if there were 5 children instead of 4?

**Tip:** When working with ratios, always simplify them to their lowest terms to make comparisons easier.

4 children shared some cards. Andy took ½ of the cards from the box first. The ratio of the number of cards Sean took from the box to the number of remaining cards after Andy took from the box was 1 : 2. The rest of cards were shared by John and Frank. The ratio of the number of cards John took to the number of the rest of cards was 2 : 3. The ratio of the number of cards Frank took to the number of cards Sean took is (Express the ratio in its simplest form).

Learn how to solve a ratio problem involving Andy, Sean, John, and Frank sharing cards. The problem includes calculating ratios and fractions to determine how many cards each child took. Suitable for Grades 6-8.

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