Question

Where do I begin with this question?

This season, Lisa's lacrosse team has won 2/3 of their home games (games played at Lisa's school), but just 2/5 of their away games (games played at other schools). In total, Lisa's team has won 26 games out of 49 games they have played. How many home games has Lisa's team played? Explain how you solved the problem.

Answer 1

Let

x = number of home games
y = number of away games

We know that they have played a total of 49 games. So that means the sum of x and y must be 49, ie

x+y = 49

Solve for either variable. I'm going to solve for y to get

y = 49 - x

Answer 2

"Lisa's lacrosse team has won 2/3 of their home games"

So (2/3)*x is the number of home games the team has won.

The team has also won "2/5 of their away games"

They won (2/5)*y away games

with me so far?

Answer 3

Yes, I am with you.

Answer 4

so in total, we have

(2/3)*x + (2/5)*y

games won

Answer 5

that total is equal to the 26 total games won

Answer 6

that means

(2/3)*x + (2/5)*y = 26

Answer 7

Now those fractions complicate things, so we can multiply both sides by the LCD 15 to clear them out.


(2/3)*x + (2/5)*y = 26

15*[ (2/3)*x + (2/5)*y ] = 15*26

15*[ (2/3)*x ] + 15*[ (2/5)*y ] = 15*26

10x + 6y = 390

Answer 8

So we have gone from

(2/3)*x + (2/5)*y = 26

to

10x + 6y = 390

Answer 9

See where to go from here?

Answer 10

Got it!

Answer 11

ok great. Tell me what you get

Answer 12

10h = 390 -6a?

Answer 13

so you let

h = # of home games
a = # of away games

just so we're on the same page

Answer 14

yes

Answer 15

we have this 10h + 6a = 390

Answer 16

previously I solved for y, which is also 'a' (two names for the same thing)

I got

y = 49 - x

which is the same as

a = 49 - h

Answer 17

You then take that and plug it into 10h + 6a = 390


10h + 6a = 390

10h + 6(49 - h) = 390 ... everywhere you see an 'a', replace it with 49-h


now solve for h

Answer 18

That substitution only works because a = 49 - h

Answer 19

10h + 294 -6h = 390

4h = 96

h=96/4

h=24

Answer 20

So, the total number of home games is 24

Answer 21

very good

Answer 22

Thank you, thank you, THANK YOU!

Answer 23

that leaves 49-h = 49-24 = 25 away games

notice how

(2/3)*h = (2/3)*24 = 16 games were won at home

(2/5)*a = (2/5)*25 = 10 games were won on the road

So 16+10 = 26 games were won total (that works out)

Answer 24

you're welcome