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We are given a problem involving animals in a zoo and their food consumption rates. Let's break this down and solve it step by step.

### Given Information:
1. **Lions and Leopards**:
   - 3 lions eat as much as 4 leopards.
   - So, 1 lion eats as much as \( \frac{4}{3} \) leopards.

2. **Bears and Leopards**:
   - 5 bears eat as much as 9 leopards.
   - So, 1 bear eats as much as \( \frac{9}{5} \) leopards.

3. **Food Available**:
   - There is enough food for 9 lions and 14 leopards to last for 30 days.

4. **Question**:
   - How long will the food last if there are 12 lions, 10 bears, and 5 leopards?

### Step 1: Express all animals in terms of leopard equivalents

1. **Lions**:
   - 1 lion = \( \frac{4}{3} \) leopards.
   - 9 lions = \( 9 \times \frac{4}{3} = 12 \) leopards.

2. **Bears**:
   - 1 bear = \( \frac{9}{5} \) leopards.
   - 10 bears = \( 10 \times \frac{9}{5} = 18 \) leopards.

So, 12 lions, 10 bears, and 5 leopards would consume as much food as:
\[
12 \times \frac{4}{3} + 10 \times \frac{9}{5} + 5 = 16 + 18 + 5 = 39 \text{ leopards}.
\]

### Step 2: Determine the total food available

The food is sufficient for:
\[
9 \text{ lions} + 14 \text{ leopards} = 12 \text{ leopards} + 14 \text{ leopards} = 26 \text{ leopards}.
\]
This food is meant to last 30 days for 26 leopard-equivalents. Therefore, the total amount of food available (in leopard-days) is:
\[
26 \times 30 = 780 \text{ leopard-days}.
\]

### Step 3: Find how long the food will last

If there are 12 lions, 10 bears, and 5 leopards, equivalent to 39 leopards, the food will last:
\[
\frac{780 \text{ leopard-days}}{39 \text{ leopard-equivalents}} = 20 \text{ days}.
\]

### Final Answer:
The food will last for **20 days**.

Would you like further details or have any other questions? Here are some related questions to explore:

1. How would the result change if 1 lion consumed the same amount as 1 bear?
2. What if there were only 7 bears instead of 10 in the zoo?
3. How much food is consumed by 1 bear in comparison to 1 lion?
4. If the number of leopards was doubled, how would the duration of food supply change?
5. Can we derive a formula for the number of days the food lasts based on the number of lions, bears, and leopards?

**Tip:** Always start by expressing quantities in a common unit when comparing different objects, like animals' food consumption rates in this problem.

In a zoo, 3 lions eat as much food as 4 leopards, while 5 bears eat as much as 9 leopards. If they have enough food for 9 lions and 14 leopards for 30 days, how long will this food last if there are 12 lions, 10 bears and 5 leopards?

Learn how to solve a zoo animal food consumption problem involving lions, bears, and leopards. The problem uses proportions and unit conversion to calculate how long a given amount of food will last for different animals. Suitable for Grades 8-10.

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