# Proportion again !!!

• Jun 12th 2008, 07:49 AM
MathLearner
Proportion again !!!
A couple of questions..

1. Four friends P,Q,R,S are working on an assignment together. They contribute to the work in the ratio 1:2:3:4 respectively. They can complete the work individually in 1,2,3,4 days respectively. If they work one after the other , how many days does it take to complete the assignment ?

2. The weights of 3 heaps of gold are in the ratio 5:6:7. By what fractions of themselves must the first two be increased so that the ratio of the weights is changed to 7:6:5 ?

ans : 24/25 , 2/5
• Jun 13th 2008, 07:35 PM
Soroban
Hello, MathLearner!

Quote:

2. The weights of 3 heaps of gold are in the ratio 5:6:7.
By what fractions of themselves must the first two be increased
so that the ratio of the weights is changed to 7:6:5 ?

Ans: $\displaystyle \frac{24}{25},\;\;\frac{2}{5}$

The original weight are: .$\displaystyle \{5a,\;6a,\;7a\}\;\;{\color{blue}[1]}$

The new weights are: .$\displaystyle \{7b,\;6b,\;5b\}\;\;{\color{blue}[2]}$

The third weight is not changed: .$\displaystyle 7a \:=\: 5b\quad\Rightarrow\quad b \:= \:\frac{7}{5}a$

Substitute into [2]: .$\displaystyle \left\{7\left(\frac{7}{5}a\right),\;6\left(\frac{7 }{5}a\right),\;5\left(\frac{7}{5}a\right)\right\} \;=\;\left\{\frac{49}{5}a,\;\frac{42}{5}a,\;7a\rig ht\}$

The first weight went from $\displaystyle 5a\text{ to }\frac{49}{5}a$, an increase of: .$\displaystyle \frac{49}{5}a -5a \:=\:\frac{24}{5}a$
. . The fraction of increase is: .$\displaystyle \frac{\frac{24}{5}a}{5a} \:=\:\boxed{\frac{24}{25}}$

The second weight went from $\displaystyle 6a$ to $\displaystyle \frac{42}{5}a$, an increase of: .$\displaystyle \frac{42}{5}a - 6a \:=\:\frac{12}{5}a$
. . The fraction of increase is: .$\displaystyle \frac{\frac{12}{5}a}{6a} \:=\:\boxed{\frac{2}{5}}$

• Jun 15th 2008, 12:59 AM
MathLearner
Thank You Sorobon !!!