Sometimes one bar model is not enough. Here’s an interesting double bar model problem using before-after relationships.
If Abby gives $10 to Ben, then they will have the same amount of money. If Ben gives $20 to Abby, then she’ll have four times as much as he has. How much money does each of them have now?
Before we start solving this problem, let’s look at our objectives, i.e. what do we need to know to solve the problem that we do not know yet. Here’s the basic bar model with the missing unknowns.
If we know 1) how much Ben has, and 2) how much more Abby has, we can know the total amount they have together.
OK, now that we know what we’re looking for, let’s get started. The easier one is the difference, i.e. how much more does Abby have. Here’s the before-after bar model.
Without any calculations, we see that in order for Abby to have the same amount as Ben after giving him $10, she must have $20 more than Ben in the beginning. Hence,
Abby has $20 more than Ben.
Now that we solved one part of the puzzle, let’s look at how we can use it to solve the remaining unknown – how much does Ben have? Let’s look at another before-after bar model.
Since Abby has $20 more than Ben (from above), if Ben were to give Abby $20, Abby would get $20 more and Ben $20 less. Looking at the bar model after the change, we can easier see the final difference after the transaction is $60.
Now, since the ratio is 4:1 between the two bars, 3 units = 60, so 1 unit = 20. Putting the $20 back, we see Ben actually had 1 unit + $20 = $40. So,
Ben has $40.
Now, putting both pieces of information together, we have
Ben has $40 and Abby has $40 + $20 = $60.
Now, just for our sanity, let’s check our numbers.
If Abby gives $10 to Ben, then they will have the same amount of money.
Abby: $60-$10 = $50,
Ben: $40+$10 = $50.
Same. checked.
If Ben gives $20 to Abby, then she’ll have four times as much as he has.
Ben: $40-$20 = $20,
Abby: $60+$20 = $80.
$80 is 4 times $20. Checked.
Yay.
