SOLUTION: There is one bus driver, two coaches, and three teams, each with four dancers going to a competition. Nia writes the expression 1 + 2 + 3 ∙ 4 to represent everyone that will be o

Question 1189804: There is one bus driver, two coaches, and three teams, each with four dancers going to a competition. Nia writes the expression 1 + 2 + 3 ∙ 4 to represent everyone that will be on the bus and solves it using the following steps:
1 + 2 + 3 ∙ 4
= 3 + 3 ∙ 4
= 6 ∙ 4
=24
A. Nia concludes that 24 people will be on the bus going to the competition. Describe the mistake Nia made when solving her expression. Then write an explanation and draw a picture to help Nia understand how to solve her expression correctly.
B. How would you write this expression if n represents the number of teams?
C. How would this expression be different if there were two coaches for each team? Write an expression and draw a picture to support your expression.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part A

Nia made the mistake in line 3
She went from 3+3*4 to 6*4 which is not correct
It's a common mistake students make at first is that they add before multiplying. Refer to PEMDAS or BODMAS or "order of operations" for more information.

Nia should have simplified the 3*4 into 12 first before adding later.
So,
3+3*4 = 3+12 = 15
There are 15 people going on the bus.

One way to draw a picture is to have 3 circles and four dots per circle. That will represent the 3*4 = 12 dancers. Then have 1 dot to represent the driver, and 2 more dots for the coaches. Count out all the dots and you should find there are 15 dots total.
The driver and coaches are not in any of the circles.

========================================================
Part B

The 3*4 means we have 3 circles of 4 dots per circle.
If we had say 5 circles, each four dots, then we'd have 5*4 = 20 dots in all the circles combined.

The question is: what if we don't know how many teams there are? Since this is unknown, we refer to it as the variable n.

The general rule is that having n circles means there are 4n dots overall in the circles alone. Simply multiply the number of dots per circle with the number of circles.

The original expression of
1+2+3*4
would turn into
1+2+n*4
which is the same as
1+2+4n
since we can multiply in any order

Then to tidy things up just a bit, the 1+2 becomes 3
So we end up with 3+4n as the expression to reprsent the total number of people going on the bus. Where n is the number of teams.

The expression 3+4n means we have 4 dots per circle (n number of circles) and 3 extra dots outside all the circles to represent the 1 driver + 2 coaches.

Answer: 3+4n

========================================================
Part C

If we had 2 coaches per team, and not just 2 coaches flat, then we'd have 2n coaches for n number of teams.

The expression
1+2+4n
would become
1+2n+4n
and that simplifies to
1+6n

Each circle now has 6 dots to represent the 4 dancers + 2 coaches per circle. Like before, n is the number of teams and it's some positive whole number (eg: n = 5). Outside all the circles is one single dot to represent the driver.

Answer: 1+6n

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