balance beam alebraic word problem

Question

on a balance beam, 3 bananas balances 4 oranges. 3 oranges balance 5 grapes and 2 grapes balance 5 tangerines. How many bananas would be needed to balance 6 tangerines?

Answer 1

I like to break problems like this into mathematical equations.

 

3 bananas balances 4 oranges

   3b=4o

 

3 oranges balance 5 grapes

   3o=5g

 

2 grapes balance 5 tangerines

   2g=5t

 

How many bananas would be needed to balance 6 tangerines?

    6t=?b

 

So, we need to find the relationship between tangerines and bananas.

 

We can use the equalities as ratios.  We know that both sides of the equation are equal (thus the equals sign), so multiplying by 3b/4o allows us to change units (or in this case, fruit) without changing the physical amount.  For example, if I weigh 110lbs, and there are 2.2lbs per 1 kg (2.2lbs=1kg), I weigh 110lbs*(1kg/2.2lbs)=55kg.  I still physically weigh the same, but the units are different.

 

So, we need to know how many bananas would be needed to balance 6 tangerines.

 

  Let's start with the 6 tangerines, and transition into bananas using ratios.  Since we start with tangerines, our ratio needs to have tangerines in the denominator to cancel out the units.  The only equation we have with tangerines is 2 grapes balance 5 tangerines

 

6t*(2g/5t) = 12/5 g = 2.4 g

 

  Now we know that 6 tangerines balances 2.4 grapes, but we are still not in bananas.  We know 3 oranges balance 5 grapes (we've already used the grapes to tangerines equation), so we'll use that one.

 

2.4g*(3o/5g) = 1.44 o

 

 So we've learned that 6 tangerines balances 2.4 grapes, which balances 1.44 oranges.  We'll use the last equation to get to bananas: 3 bananas balances 4 oranges.

 

1.44o*(3b/4o)=1.08 bananas

 

  6 tangerines therefore balances with 1.08 bananas.