Caged counting

Note that each can be written as an arithmetic sequence with the form $t(n)=an+b$ .

With a starting value of 502 and increasing by 2, the equation is $t(n)=2n+502$

The other has a starting value of 999 and goes down by 5. This gives us $t(n)=-5x+999$

Since we want them to count the same number, we want $t(n)$ to be equal. So we can substitute one equation into the other.

$2x+502=999-5x$

$7x=497$

$x=71$

Plugging in $x$ into one of the equations

$t(x)=2(71)+502$

$t(x)=142+502$

$t(x)=\boxed{644}$

502 + 2x = 999 - 5x

let x = the number of times 2 & 5 will be added and subtracted respectively, y = the number which Hansel and Gretel say at the same time

Given these two variables, simultaneous equations may be used.

Hansel: 502 + 2x = y

Gretel: 999 - 5x = y

Rearrange and add the two equations:

2x - y = -502

5x + y = 999

7x = 497

    x = 71

Now that the value of x is obtained, we can get the value of y by substituting to either of the 2 original equations:

502 + 2(71) = y

 y = 644

BY ARITHMETIC PROGRESSION,nth TERM WII BE, 502+[n-1]2=999+[n-1] [-5] BY SOLVING, WE GET n=72 SUBSTITUTING, 502+71 2=644

its like physics

where displacement=first position+/- velocity times time

in this case displacement1=displacement2

hansel's velocity= 2 number/sec gretel's velocity=-5number/sec

p1+v1t=p2-v2t

502+2t=999-5t solve the equation you'll get t=71 second

insert t=71 to d=502+2t or d=999-5t you'll get d=644

It is arithmetic sequence.a is first term and d is tolerance.let An is the number they will say at the same time.Hansel:a=502 d=2.Gretel:a=999 d=-5[nagative because is backward] using the formula. An=a+[n-1][d] Substitute a and d into the formula.502+[n-1][2]=999+[n-1][-5] .Hence,n=72.Substitute n=72 by using a=502,d=2 or a=999,d=-5 can get An=644 \boxed{644}

nth term of first AP is equal to the nth term of second AP. By solving, we get n as 72. So the 72th term of both the APs will be the required number i.e. 644

This problem could be thought of a simple time and distance based problem where two people on the opposite ends of a town are moving towards each other, the towns being the numbers 502 and 999 respectively and the distance between them being 497. The number that they say together is analogous to the time when they meet but since both of them start at the same time, so every time they meet their distances are functions of their speed and here that ratio is 2:5. Thus the total distance is to be divided in 2:5 or 2/7th of 497 for hansel and 5/7th of 497 for gretel.

We can assume that the count one number each second. Changing this rate will only affect the time after which they count that same number, not the number itself.

So we have 999-5 1(num/sec) t(in secs)=502+2 1 t

This gives t=71 secs, ie they will count the same no. after 71 seconds.

thus the number is 502+2 1 71=644

Ok.

That's how I approached this problem H has to count 5 numbers to break the 10 barrier while G only needs two numbers and since they count at the same pace, that means 1 of Gretel's + 1 of Hansel's downs the total value of the deference between them by 7

(G=5)+(H=2)=7

g=5/7, h=2/7, where g & h are the ratios so the needed number N will be $N_H=h×(999-502)+502=142+502=644$

or

$N_G=999-g×(999-502)=999-355=644$

And you can simply check that they both counted the same number of numbers by doing this

$n_H=2/7×497/2=1/7×497=71$

$n_G=5/7×497/5=1/7×497=71$

It's like relative motion

502 + 2x = 999 - 5x

The following condition is to be met when they have to say the same number. 502 + 2 n = 999 - 5 n. Solving the above, we get n = 71. Substituting this value in the above eqn, we ger 644 as an answer.

using ap a+(n-1)d=b+(n-1)e 502+(n-1)2=999+(n-1)-5 n=72 no.= 502+71*2=644

502 + n * 2 = 999 - 5 * n

n = 71

the magic number is 502 + 71 * 2 = 644

Let this number = x

Hansel --> 502 + 2n = x Gretal --> 999 - 5n = x

502 + 2n = 999 - 5n 2n + 5n = 999 - 502 7n = 497 n = 71

x = 2*71 + 502 = 644

check: x = 999 - 5*71 = 644

2n+502 = -5n+999 2n+5n = 999-502 7n = 497 n = 71

use excel functions one column for 502 and (=502+2), one clomun for 999 and (=999-5), find the row that has 2 same numbers. =)