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Grade 7/8 POTW First, 15 girls left g - 15 There were now double the amount of boys as girls. 2(g - 15) = b 2g - 30 = b 3/4 of boys left while 1/3 of girls left. 2/3(g - 15) = 1/4b There are now 14 more boys than girls. 2/3(g - 15) = 1/4b + 14 Now for the actual math. We first need to find a common multiple between 3 and 4 to make 2/3 and 1/3 whole numbers. 3 x 4 = 12 2 x 12 = 24 24/3 = 8 1 x 12 = 12 12/4 = 3 2/3(g - 15) = 1/4b + 14 8(g - 15) = 3b + 14 8g - 120 = 3b + 168 8g - 120 + 120 = 3b + 168 + 120 We can now use substitution. 8g = 3(2g - 30) + 288 8g = 6g - 90 + 288 8g = 6g + 198 8g = 6g + 198 - 6g 2g = 198 2g/2 = 198/2 g = 99 Now to use this value in the previous equation. 2g - 30 = b 2(99) - 30 = b 198 - 30 = b 168 = b So it would be: 168 + (99 - 15) 168 + 84 1/4(168) + 2/3(84) 42 + 56 98 students remained to perform in the school musical.
Ok, no time to explain, I'm just going to do the algebra. Let x be the number of boys who came to the meeting, Let y be the number of girls who came to the meeting. (Yay I actually used two variables)
x = 2(y-15) Thus x = 2y-30
Equation: 1/4x + 14 = 2/3(y-15) This is literally a translation of the sentence stating that a bunch of people left.
I'm not going to write the check, but my answer is correct after checking. Overall, there started off with 168 boys and 84 girls. By using the check, you can run through everything and determine that 42 boys remained, and 56 girls remained, a total of 98 people. -Alan
POTW: Why do we still have to do POTW over the break? (Kidding. Kind of)
b=boys g=girls t=total
t-15= 3b (2b=t-15). 15 girls left, which left the girls (2b) and the boys (b).
t-15= 3/4b + 1/3g. 15 more girls left, which left 3/4 boys and 1/3 girls left.
g=b+14. This left 14 more girls than boys, meaning girls is b+14. The total would be b+14+b= 2b+14.
(I don't like this already (kidding.. I think)). Anyways, to start this POTW, I need to find out the value of b (since the end is 2b + 14). The second point says that 15 girls left, leaving 1/3 girls. Then meant that 2/3 of the girls left, meaning that 15 is 2/3 of the girls. If you divide this by 2, you will get 1/3 (7.5), then you can multiply that by 3 to get the total "girl population", which is 22.5. Since girls is b+14, we can subtract 14 from 22.5 to find the "boy population", which is 8.5. To find the number of people that went on stage, we can add b+b+14, which is: 2b+14 =8.5+8.5+14 =17+14 =31. 31 students went on stage.
let me start off by saying this question made no sense to me. I didn't understand the "later after the 15 girls left, 3/4 of the boys and 1/3 of the remaining left.. or something like that. so, here I go. let y be girls let x be boys
first equation: 2(y - 15) = x 2y - 30 = x
second equation:
1/4x + 14 = 2/3(y - 15) 1/4(2y - 30) + 14 = 2/3y - 10 1/2y - 7.5 + 14 = 2/3y - 10 1/2y - 6.5 = 2/3y - 10 1/2y = 2/3y - 3.5 4/6y - 3/6y = 3.5 1/6y = 3.5 y = 21 x = 21(2) - 30 x = 12 I'm pretty sure my answer is wrong since I put it back into the question and didn't work. Well, at least I tried...
First 15 girls left and 2 times as many girls were remaining. 2(g-15) = b 2g-30 = b
If I were to translate all of the people leaving into an algebraic equation, I would get: 2/3(g-15) = 1/4b + 14
I can substitute b with: b = 2g-30 2/3(g-15) = 1/4(2g-30) + 14 2/3g - 10 = 0.5g - 7.5 + 14 2/3g - 10 = 1/2g + 6.5 2/3g = 1/2g + 16.5 4/6g = 3/6g + 16.5 1/6g = 16.5 g = 99
There were 99 girls to begin with Now to plug this in to my first equation: b = 2g-30 b = 2(99)-30 b = 198-30 b = 168 There were 168 boys to begin with.
If I were to start with these numbers and subtract everyone who left: 168 + 99 168 + 84 <---(99-15) 1/4*168 + 2/3*84 42 + 56 98
There are 98 people in the school production. Check is done on paper.
Let: B = boys g = girls t = total Once 15 girls left, there is 2 times the amount of boys as girls. 3/4 of boys and 1/3 of girls left. Left 14 more girls than boys g - 15 = 2g The people who left in an equation would be 2/3(g-15) = 1/4 b + 14 2/3(g-15)=1/4 b-30+14 Using common multiples for 3 and 4, you can make 2/3 and 1/4 equal. 8(g-15) = 3b+14 8g - 120 = 3b + 168 8g = 3b + 168 + 120 8g = 3b + 288 Substitute b for 2g - 30 8g = 3(2g - 30) + 288 8g = 6g - 90 + 288 8g = 6g + 198 2g = 198 2g/2 = 198/2 g = 98 98 boys and girls stayed to perform.
Info: Early Morning - 15 girls left - twice as many boys as girls remained Later - 3/4 of the boys and 1/3 of the remaining girls - 14 more girls than boys Question: How many students remained to perform in the school musical?
ALRIGHT. Let g = girls, and let b = boys. First, we know that 15 girls left, so: g - 15 Now there are double the number of boys, so: 2(g - 15) = b The above equation can be simplified to: 2g - 30 = b
Moving on... 3/4 of the boys left, while 1/3 of the girls left. So now: 2/3(g-15) = 1/4b We also know there are 14 more boys. 2/3(g-15) = 1/4b + 14
TIME FOR THE ACTUAL SOLVING!! :DDD The equation we ended up with was this: 2/3(g-15) = 1/4b + 14
Now to make 2/3 or 1/4 equal, simply find the common multiples of both 3 and 4. The equation now looks like this: 8(g-15) = 3b+14 8g - 120 = 3b + 168 8g = 3b + 168 + 120 8g = 3b + 288
Next we need to substitute b (hahhaahahaha) We can do this by instead of using "b", using (2g-30) 8g = 3(2g - 30) + 288 8g = 6g - 90 + 288 8g = 6g + 198 2g = 198 2g/2 = 198/2 g = 98
98 boys and girls stayed to perform. I'm not going to find out how many boys and how many girls individually, because it didn't ask that.
I did the POTW on paper, at first there are 168 boys and 84 girls. In the end, there were only 42 boys and 56 girls. So 98 people performed at the musical. I wonder why they didn't just stick around for the whole meeting then not perform?
POTW #23: Time to catch up! (Sorry for the messiness in my explanation)
After 3 tries and miscalculations, I finally was able to solve this. A total of 98 students remained to perform in the school musical. I would put all my work down, but it's quite messy and clustered with equations. The main one I used to solve for how many girls there were was: let g = # of girls in the beginning let b = # of boys in the beginning important expressions/equations g-15 b=2g-30 (simplified ver. of b=2(g-15) 2/3(g-15) = 1/4b + 14 I then substituted b for the equation that included the variable g. I divided that by 4 since it was originally 1/4b. 2/3(g-15)=((2g-30)/4)+14 That then lead up to me finding that there were 99 girls in the beginning. From there on I followed the steps/pieces of information. Therefore, 98 students performed the musical.
Grade 7/8 POTW
ReplyDeleteFirst, 15 girls left
g - 15
There were now double the amount of boys as girls.
2(g - 15) = b
2g - 30 = b
3/4 of boys left while 1/3 of girls left.
2/3(g - 15) = 1/4b
There are now 14 more boys than girls.
2/3(g - 15) = 1/4b + 14
Now for the actual math. We first need to find a common multiple between 3 and 4 to make 2/3 and 1/3 whole numbers.
3 x 4 = 12
2 x 12 = 24
24/3 = 8
1 x 12 = 12
12/4 = 3
2/3(g - 15) = 1/4b + 14
8(g - 15) = 3b + 14
8g - 120 = 3b + 168
8g - 120 + 120 = 3b + 168 + 120
We can now use substitution.
8g = 3(2g - 30) + 288
8g = 6g - 90 + 288
8g = 6g + 198
8g = 6g + 198 - 6g
2g = 198
2g/2 = 198/2
g = 99
Now to use this value in the previous equation.
2g - 30 = b
2(99) - 30 = b
198 - 30 = b
168 = b
So it would be:
168 + (99 - 15)
168 + 84
1/4(168) + 2/3(84)
42 + 56
98 students remained to perform in the school musical.
Ok, no time to explain, I'm just going to do the algebra.
ReplyDeleteLet x be the number of boys who came to the meeting, Let y be the number of girls who came to the meeting.
(Yay I actually used two variables)
x = 2(y-15)
Thus x = 2y-30
Equation:
1/4x + 14 = 2/3(y-15)
This is literally a translation of the sentence stating that a bunch of people left.
Solve: 1/2y - 7.5 + 24 = 2/3y
1/2y + 16.5 = 2/3y
2/3y - 1/2y = 16.5
4/6y-3/6y = 16.5
1/6y = 16.5
y = 99
x = 2*99-30
x = 168
I'm not going to write the check, but my answer is correct after checking.
Overall, there started off with 168 boys and 84 girls. By using the check, you can run through everything and determine that 42 boys remained, and 56 girls remained, a total of 98 people.
-Alan
POTW:
ReplyDeleteWhy do we still have to do POTW over the break? (Kidding. Kind of)
b=boys
g=girls
t=total
t-15= 3b (2b=t-15). 15 girls left, which left the girls (2b) and the boys (b).
t-15= 3/4b + 1/3g. 15 more girls left, which left 3/4 boys and 1/3 girls left.
g=b+14. This left 14 more girls than boys, meaning girls is b+14. The total would be b+14+b= 2b+14.
(I don't like this already (kidding.. I think)). Anyways, to start this POTW, I need to find out the value of b (since the end is 2b + 14). The second point says that 15 girls left, leaving 1/3 girls. Then meant that 2/3 of the girls left, meaning that 15 is 2/3 of the girls. If you divide this by 2, you will get 1/3 (7.5), then you can multiply that by 3 to get the total "girl population", which is 22.5. Since girls is b+14, we can subtract 14 from 22.5 to find the "boy population", which is 8.5. To find the number of people that went on stage, we can add b+b+14, which is:
2b+14
=8.5+8.5+14
=17+14
=31.
31 students went on stage.
Or not, I guess.
Deletelet me start off by saying this question made no sense to me. I didn't understand the "later after the 15 girls left, 3/4 of the boys and 1/3 of the remaining left.. or something like that.
ReplyDeleteso, here I go.
let y be girls
let x be boys
first equation:
2(y - 15) = x
2y - 30 = x
second equation:
1/4x + 14 = 2/3(y - 15)
1/4(2y - 30) + 14 = 2/3y - 10
1/2y - 7.5 + 14 = 2/3y - 10
1/2y - 6.5 = 2/3y - 10
1/2y = 2/3y - 3.5
4/6y - 3/6y = 3.5
1/6y = 3.5
y = 21
x = 21(2) - 30
x = 12
I'm pretty sure my answer is wrong since I put it back into the question and didn't work. Well, at least I tried...
let g = girls
ReplyDeletelet b = boys
let t = total
First 15 girls left and 2 times as many girls were remaining.
2(g-15) = b
2g-30 = b
If I were to translate all of the people leaving into an algebraic equation, I would get:
2/3(g-15) = 1/4b + 14
I can substitute b with: b = 2g-30
2/3(g-15) = 1/4(2g-30) + 14
2/3g - 10 = 0.5g - 7.5 + 14
2/3g - 10 = 1/2g + 6.5
2/3g = 1/2g + 16.5
4/6g = 3/6g + 16.5
1/6g = 16.5
g = 99
There were 99 girls to begin with
Now to plug this in to my first equation:
b = 2g-30
b = 2(99)-30
b = 198-30
b = 168
There were 168 boys to begin with.
If I were to start with these numbers and subtract everyone who left:
168 + 99
168 + 84 <---(99-15)
1/4*168 + 2/3*84
42 + 56
98
There are 98 people in the school production. Check is done on paper.
Grade 8 POTW
ReplyDelete56 girls and 42 boys were left to star in the school's musical. I did my work in my math notebook.
So what's the final answer to the question asked?
DeleteLet:
ReplyDeleteB = boys
g = girls
t = total
Once 15 girls left, there is 2 times the amount of boys as girls.
3/4 of boys and 1/3 of girls left.
Left 14 more girls than boys
g - 15 = 2g
The people who left in an equation would be 2/3(g-15) = 1/4 b + 14
2/3(g-15)=1/4 b-30+14
Using common multiples for 3 and 4, you can make 2/3 and 1/4 equal.
8(g-15) = 3b+14
8g - 120 = 3b + 168
8g = 3b + 168 + 120
8g = 3b + 288
Substitute b for 2g - 30
8g = 3(2g - 30) + 288
8g = 6g - 90 + 288
8g = 6g + 198
2g = 198
2g/2 = 198/2
g = 98
98 boys and girls stayed to perform.
98 people performed in the musical, unlike those that ditched them!
ReplyDeleteDitchers!
DeleteGrade 8 POTW:
ReplyDeleteVIsit the document for my answer
https://docs.google.com/document/d/1Q6eTTob9ozzsX21xtl0wAgE6EMXTuuAXu2U9diol4KA/edit
POTW #23:
ReplyDeleteInfo:
Early Morning
- 15 girls left
- twice as many boys as girls remained
Later
- 3/4 of the boys and 1/3 of the remaining girls
- 14 more girls than boys
Question:
How many students remained to perform in the school musical?
ALRIGHT.
Let g = girls, and let b = boys.
First, we know that 15 girls left, so: g - 15
Now there are double the number of boys, so: 2(g - 15) = b
The above equation can be simplified to: 2g - 30 = b
Moving on...
3/4 of the boys left, while 1/3 of the girls left.
So now: 2/3(g-15) = 1/4b
We also know there are 14 more boys.
2/3(g-15) = 1/4b + 14
TIME FOR THE ACTUAL SOLVING!! :DDD
The equation we ended up with was this:
2/3(g-15) = 1/4b + 14
Now to make 2/3 or 1/4 equal, simply find the common multiples of both 3 and 4.
The equation now looks like this:
8(g-15) = 3b+14
8g - 120 = 3b + 168
8g = 3b + 168 + 120
8g = 3b + 288
Next we need to substitute b (hahhaahahaha)
We can do this by instead of using "b", using (2g-30)
8g = 3(2g - 30) + 288
8g = 6g - 90 + 288
8g = 6g + 198
2g = 198
2g/2 = 198/2
g = 98
98 boys and girls stayed to perform. I'm not going to find out how many boys and how many girls individually, because it didn't ask that.
(Check done on paper.)
I did the POTW on paper, at first there are 168 boys and 84 girls. In the end, there were only 42 boys and 56 girls. So 98 people performed at the musical. I wonder why they didn't just stick around for the whole meeting then not perform?
ReplyDeletePOTW #23: Time to catch up! (Sorry for the messiness in my explanation)
ReplyDeleteAfter 3 tries and miscalculations, I finally was able to solve this. A total of 98 students remained to perform in the school musical. I would put all my work down, but it's quite messy and clustered with equations.
The main one I used to solve for how many girls there were was:
let g = # of girls in the beginning
let b = # of boys in the beginning
important expressions/equations
g-15
b=2g-30 (simplified ver. of b=2(g-15)
2/3(g-15) = 1/4b + 14
I then substituted b for the equation that included the variable g. I divided that by 4 since it was originally 1/4b.
2/3(g-15)=((2g-30)/4)+14
That then lead up to me finding that there were 99 girls in the beginning. From there on I followed the steps/pieces of information.
Therefore, 98 students performed the musical.