1. ## Easy Math :P

Should be easy for you guys anyways... Okay...

1)

Simplify

p3r/pq X q2/pr2

2)

Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount G poured into the larger container. (Is this possible? Don't give you any real information)

3) 30% of the people in a restaurant are adults while the rest are children. Three-fifths of the adults are men and 3/7 of the children are boys. There are 112 more girls than women. How many people are there in the restaurant?

4) The ratio of the weight of a cow to that of a goat is 8:3. The ratio of the weight of the goat to that of a sheep is 5:4. The cow is heavier than the sheep by 112 kg. Find the total weight of the three animals.

2. Originally Posted by Rocher

2)

Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount G poured into the larger container. (Is this possible? Don't give you any real information)
The small container gets: $\displaystyle 20-G$
3) 30% of the people in a restaurant are adults while the rest are children. Three-fifths of the adults are men and 3/7 of the children are boys. There are 112 more girls than women. How many people are there in the restaurant?
$\displaystyle \frac{3}{10}$ are adults so $\displaystyle \frac{7}{10}$ are children.

$\displaystyle \frac{3}{7}$ of the children are boys, so $\displaystyle \frac{7}{10}\times\frac{3}{7}=\frac{3}{10}$ of the group is boys.

That means that: $\displaystyle \frac{7}{10}-\frac{3}{10}=\frac{4}{10}=\frac{2}{5}$ of the group are girls (because all the children that aren't boys are girls.... hopefully)

Anyway, $\displaystyle \frac{3}{5}$ of the adults are men, so then $\displaystyle \frac{3}{10}\times\frac{3}{5}=\frac{9}{50}$ of the group are men.

That means that $\displaystyle \frac{3}{10}-\frac{9}{50}=\frac{6}{50}=\frac{3}{25}$ of the group are women.

So we know that ($\displaystyle g$ stands for the number of people in the group): $\displaystyle g\times\frac{2}{5}-g\times\frac{3}{25}=112$

So: $\displaystyle g\left(\frac{2}{5}-\frac{3}{25}\right)=112$

Subtract: $\displaystyle g\left(\frac{7}{25}\right)=112$

divide both sides by $\displaystyle \frac{7}{25}$ to get: $\displaystyle \boxed{g=400}$

3. Thank you, now just need 1 and 4 :P

4. The key to the fouth problem is to write it as fractions

cows/goat 8/3 goat/sheep 5/4

and then find the lcm of the goat weight.

cows/goat 40/15 goat/sheep 15/12

and then reason this:

If the ratio of the cow's weight to the goat's weight is 40:15 and the goat's to the sheep's weight is 15:12, therefore the the cow's to sheep's ratio is 40:12. Then we have the given ratio of cow to sheep equals the ratio of the cow (which is x + 112) and the sheep (x)

40/12 = (x + 112)/x

The rest is algebra. if you have any problems just ask.

5. For the first one:

a/b * c/d = ac/bd

and

Anything over itself is one so b/b = 1

e.g. a3/a = a/a * 3 = 1 * 3 = 3

6. Originally Posted by Rocher
Should be easy for you guys anyways... Okay...

1)

Simplify

p3r/pq X q2/pr2

2)

Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount G poured into the larger container. (Is this possible? Don't give you any real information)

3) 30% of the people in a restaurant are adults while the rest are children. Three-fifths of the adults are men and 3/7 of the children are boys. There are 112 more girls than women. How many people are there in the restaurant?

4) The ratio of the weight of a cow to that of a goat is 8:3. The ratio of the weight of the goat to that of a sheep is 5:4. The cow is heavier than the sheep by 112 kg. Find the total weight of the three animals.
Hey, I love easy Math!
And I hate more than one problem in one posting.
But I have lots of sparetime today.
So, what the heck....

1) p3r/pq X q2/pr2
If that is [(p^3)r]/(pq) *[(q^2)/(p*r^2)]
Then,
= [(numerator 1)(numerator 2)] / [(denominator 1)(denominator 2)]
= [(p^3)(q^2)(r)] / [(p^2)(q)(r^2)]
= p^(3-2) * q^(2-1) * r^(1-2)
= p^(1) *q^(1) * r^(-1)

.........................................

2) If G gals of the 20 gals is poured into the large container, then (20 -G) gals is poured into the small container.

.................................................. .....................................

3) Let P = number of people; A = number of adults; C = number of chlidren; M = number of men; W = number of women; B = number of boys; G = number of girls
P = A +C ------(i)
A = M +W ----(ii)
C = B +G -----(iii)

"30% of the people in a restaurant are adults while the rest are children..."
---0.3P = A
---0.7P = C

"...Three-fifths of the adults are men..."
----(3/5)A = M
so, (3/5)*(0.3P) = M
hence, M = 0.18P
And, W = 0.3P -0.18P = 0.12P ----(1)

"...and 3/7 of the children are boys."
----(3/7)C = B
so, (3/7)*(0.7P) = B
hence, B = 0.3P
And, G = 0.7P -0.3P = 0.4P ----(2)

"There are 112 more girls than women..."
----G = W +112
Plugging in those from (1) and (2),
0.4P = 0.12P +112
0.4P -0.12P = 112
0.28P = 112
P = 112/(0.28) = 400 people in the restaurant. --------answer.

.................................................. ...............

4) Let C = weight of cow; G = weight of goat; S = weight of sheep.

"The ratio of the weight of a cow to that of a goat is 8:3."
---C/G = 8/3
Cross multiply,
3C = 8G ----(1)

"The ratio of the weight of the goat to that of a sheep is 5:4."
---G/S = 5/4
Cross multiply,
4G = 5S ----(2)

"The cow is heavier than the sheep by 112 kg."
C = S +112 ----(3)

[I wonder why the number 112 is popular in your set of questions?]

"Find the total weight of the three animals."
Okay, there are many ways to continue this. One of them is to express the C and S in terms of G so that in Eq.(3) there will be only one variable, the G.

From (1), C = (8G)/3
From (2), S = (4G)/5
Substitute those into (3),
(8G)/3 = (4G)/5 +112
Clear the fractions, multiply both sides by 3*5,
5(8G) = 3(4G) +(3*5)(112)
40G = 12G + 1680
40G -12G = 1680
G = 1680/28 = 60 kg

And so,
C = (8G)/3 = (8*60)/3 = 160 kg
S = (4G)/5 = (4*60)/5 = 48 kg

Therefore, C+G+S = 160+60+48 = 268 kg total weight. ------answer.