Solving for Sunny Days: An Unusual Banana Schedule

Imagine a curious monkey with a strict banana eating schedule. This monkey consumes a varying number of bananas depending on the weather: 20 bananas on a sunny day and 12 bananas on a rainy day. Over a period of 8 consecutive days, the monkey eats a total of 112 bananas. The question is: How many sunny days are there in these 8 days?

Formulating the Problem

Let's use some algebra to solve this problem. If we denote the number of sunny days as ( s ) and the number of rainy days as ( r ), we can set up the following equations:

Equation 1:

[ s r 8 ]

This equation describes the total number of days, which is 8 in this case.

Equation 2:

[ 20s 12r 112 ]

This equation describes the total number of bananas eaten, with the monkey consuming 20 bananas on each sunny day and 12 bananas on each rainy day.

Solving the Equations Step-by-Step

Let's solve these equations step by step:

Step 1: Express ( r ) in terms of ( s ).

[ r 8 - s ]

Step 2: Substitute ( r ) in the second equation.

[ 20s 12(8 - s) 112 ]

Step 3: Expand and combine like terms.

[ 20s 96 - 12s 112 ] [ 8s 96 112 ]

Step 4: Solve for ( s ).

[ 8s 16 ] [ s 2 ]

Substituting ( s 2 ) back into the equation for ( r ):

[ r 8 - 2 6 ]

Therefore, there are 2 sunny days and 6 rainy days in these 8 consecutive days.

Verifying the Solution

To verify, let's recalculate the total number of bananas eaten over 8 days:

- 2 sunny days: ( 2 times 20 40 ) bananas - 6 rainy days: ( 6 times 12 72 ) bananas - Total: ( 40 72 112 ) bananas

This confirms that our solution is correct. The monkey indeed eats 112 bananas when there are 2 sunny days and 6 rainy days.

Alternative Methods

There are alternative methods to solve this problem:

Method 1:

If the monkey eats a minimum of 12 bananas each day, the total for 8 days is 96 bananas. The excess bananas eaten are ( 112 - 96 16 ) bananas. Since the monkey eats 8 more bananas on each sunny day, the number of sunny days is ( frac{16}{8} 2 ).

Method 2:

Let ( x ) number of sunny days and ( y ) number of rainy days. Then:

[ 2x 12y 112 ]

Multiply the first equation by 12:

[ 12x 12y 96 ]

Subtract the second equation from the first:

[ 2 - 12x 112 - 96 ] [ 8x 16 ] [ x 2 ]

Thus, the number of sunny days is 2, and the number of rainy days is 6.

Conclusion

The problem of determining the number of sunny days is a neat application of algebraic equations. By solving these equations, we can determine that there are 2 sunny days and 6 rainy days over 8 consecutive days in this banana-eating scenario.