Dimensions-Rectangular cuboid box-rejoinder

This is with reference to my earlier blog where the dimensions of a box are x y and z and these satisfy 3 equations and I wanted you to find the volume of the box.
Multiplying both the right and left hand sides of the 3 equations we get
xyz+x+y+z+1/x+1/y+1/z+1/xyz=4*7/3*1=28/3
But adding both sides of the equations we get x+1/x+y+1/y+z+1/z=4+7/3+1=22/3
Deducting result 2 from result1 we get xyz+1/xyz=2
This will be possible only when xyz=1 and so xyz the volume of the box is 1  

Dimensions-rectangular cuboid box

The dimensions of a rectangular cuboid box are x ,y and z. They satisfy the following equations-
x+(1/y)=4....y+(1/z)=1....z+(1/x)=7/3.
Find the volume of the box 

Puzzle-Syrup water mixture-Rejoinder

This refers to my previous blog wherein I had posted a question-From a container containing a syrup-water mixture in the ratio 5/3,how much should be removed and replaced by an equal quantity of water so that the syrup and water are contained equally
If  the syrup content is to be changed from 5/8 to 1/2 we have to remove 1/8 content of syrup which will need 1/5 of the total mixture--{5/8}-{(1/5)*(5/8)}=5/8-1/8=1/2.
The water content then will become 3/8-(1/5ths of 3/8)+1/5=3/8-3/40+1/5=    that is (15-3+8)/40=20/40=1/2 as required.
Other variations  ratio 5/7..1/7ths of the mixture to be replaced
                                    5/9..2/9 ths of the mixture to be replaced
                                    5/11..3/11 ths of the mixture to be replaced as follows
syrup 7/12-1/7 ths of 7/12=1/2  ..water   5/12-(5/12*1/7)+1/7=(35-5+12)/84=42/84=1/2
         9/14-2/9 ths of 9/14=1/2                 5/14-(5/14*2/9)+2/9=(45-10+28)/126=63/126=1/2
         11/16-3/11ths of 11/16=1/2            5/16-(5/16*3/11)+3/11=(55-15+48)/176=88/176=1/2

Puzzle-Syrup,water mixture

A vessel contains a liquid 3 parts of which are water  and 5 parts syrup.How much of the liquid must be drawn off and replaced with water so that the liquid may be half water and half syrup?
Alternately if the original liquid contains 5 parts water and 7 parts syrup?.

Sequence puzzle-rejoinder

This is a rejoinder to my blog posted on 11 june.-reproduced below-
A sequence of numbers a(1),a(2),a(3)......a(100) has the property that for every integer k between 1 and k inclusive the number a(k) is k less than the sum of the other 99 numbers.Find the 50th number-
There has been no response so far.
The solution can be arrived at as follows-
If S is the sum of the 100 numbers,a(k)=S-a(k)-k.
So 2a(k)=S-k
Substituting the values 1 to 100 for k  in this equation and adding all the results we will have
2S=100S-(the sum of the 100 numbers from 1 to 100) or
98S=5050 or S= 5050/98.
So a(50)=(5050/98)-a(50)-50
or 2a(50)=(5050/98)-50=150/98
or a(50)=75/98

Sequence puzzle

A sequence of numbers a1,a2,a3,....... a100 has the property that for every integer k between 1 and k inclusive,the number ak is k less than the sum of the other 99 numbers.Find the 50th number of the sequence.

Number in different forms

A person chooses a 3 digit number of the form N=ABC where A,B,C  are distinct digits.He then considered all possible numbers of the form ACB,BCA,BAC,CAB and CBA.He then added these 5 numbers  and got the result  3194.Determine the number N