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?

Let's first define the variables based on the problem:

- Let the amount of food a single leopard consumes in one day be denoted as \( f_L \) (amount of food for one leopard per day).
- The problem gives us these relationships:
  - 3 lions eat as much as 4 leopards, meaning the food consumption rate of one lion \( f_{\text{lion}} \) is \(\frac{4}{3}\) that of a leopard, i.e.:
    \[
    f_{\text{lion}} = \frac{4}{3} f_L
    \]
  - 5 bears eat as much as 9 leopards, meaning the food consumption rate of one bear \( f_{\text{bear}} \) is \(\frac{9}{5}\) that of a leopard, i.e.:
    \[
    f_{\text{bear}} = \frac{9}{5} f_L
    \]

Now, we are told that there is enough food for 9 lions and 14 leopards for 30 days. Let's calculate the total food supply based on this.

### Total food for 9 lions and 14 leopards for 30 days:

- Food consumed by 9 lions in one day:
  \[
  9 \times f_{\text{lion}} = 9 \times \frac{4}{3} f_L = 12 f_L
  \]
- Food consumed by 14 leopards in one day:
  \[
  14 \times f_L = 14 f_L
  \]
- Total food consumed per day by 9 lions and 14 leopards:
  \[
  12 f_L + 14 f_L = 26 f_L
  \]
Thus, for 30 days, the total amount of food is:
\[
30 \times 26 f_L = 780 f_L
\]
So, the total food available is equivalent to the amount consumed by 780 leopards over one day.

### Total food consumption for 12 lions, 10 bears, and 5 leopards:
Now, let's calculate how much food 12 lions, 10 bears, and 5 leopards consume in one day.

- Food consumed by 12 lions in one day:
  \[
  12 \times f_{\text{lion}} = 12 \times \frac{4}{3} f_L = 16 f_L
  \]
- Food consumed by 10 bears in one day:
  \[
  10 \times f_{\text{bear}} = 10 \times \frac{9}{5} f_L = 18 f_L
  \]
- Food consumed by 5 leopards in one day:
  \[
  5 \times f_L = 5 f_L
  \]

Total food consumed per day by 12 lions, 10 bears, and 5 leopards:
\[
16 f_L + 18 f_L + 5 f_L = 39 f_L
\]

### Time the food will last:
The total food available is equivalent to 780 leopards' daily food supply, and the total consumption per day is equivalent to 39 leopards' daily food supply. Therefore, the food will last for:
\[
\frac{780 f_L}{39 f_L} = 20 \text{ days}
\]

Thus, the food will last for 20 days.

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

1. What if the zoo had 15 lions, 8 bears, and 6 leopards?
2. How long would the food last if one lion ate twice as much as a leopard?
3. What if the bears' food consumption doubled?
4. How does the food duration change if the number of bears increases to 12?
5. How much more food would be needed to last for 30 days with the current number of animals?

**Tip:** Always break down word problems into smaller parts to simplify calculations.

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?

Find out how long the food will last for 12 lions, 10 bears, and 5 leopards in a zoo, given that 3 lions eat as much food as 4 leopards and 5 bears eat as much as 9 leopards. Explore ratios, proportions, and algebra in this detailed solution, suitable for students in Grades 8-10.

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