The average of 6 numbers increases by 2.

So the average now = 100 + 2 = 102

Then, the average of 7 numbers increases by 2 again.

So the average now = 102 + 2 = 104


The 7th number = total of 7 numbers - total of first 6 numbers

= 7 × 104 - 6 × 102

= 7 x 2 + 7 x 102 - 6 x 102

= 14 + 102

= 116

Alternatively,

When adding the 6th number,

Getting the new average is akin to having the 6th number 'donate' some of its value to the first 5 numbers.

Since the average increased by 2, each of the original 5 numbers received 2 from the 6th number.

The 6th number now decreases to 102. And 102 = 100 + 2


Total value donated by the 6th number
= 2 × 5 = 10


So the value of the 6th number
= 100 + 2 + 2 × 5 = 100 + 2 + 10 = 112


(Or 100 + 2 × 6 = 100 + 12 = 112)

Thank J

... but

Why last few step . 7x 2 +7 ? How to get ?

You can do this way...
Total for 5 number = 5×100 = 500
Total for 6 number = 6×102 = 612
Total for 7 number = 7×104 = 728
Therefore, 7th number = 728-612 = 116.

Likewise,

When adding the 7th number,

Getting the new average is akin to having the 7th number 'donate' some of its value to the first 6 numbers.

Since the average increased by 2, each of the original 6 numbers received 2 from the 7th number.

The 7th number now decreases to 104 And 104 = 102 + 2


Total value donated by the 7th number
= 2 × 6 = 12


So the value of the 7th number
= 102 + 2 + 2 × 6 = 102 + 2 + 12= 116


(Or 102 + 2 × 7 = 102 + 14 = 116)

Thank AC Lim . This method is so easy to understand too. Thanks .

Can you help me with this too.
The average of 7 different odd number is 53 . If the smallest number is 45 and the largest number is 69 . What is the middle number?

I only know 1st step as 53x7=371 . And is it minus 45 &69 then the rest divided to 5?

Thank J for your kind and detail explanation. Maybe I'm not that good at math , that why I really cannot fully understand. So sorry. I have wasted your time and effort. I'm so sad about that. I have tried hard to understand it. But will not very clear at some part sometime . But I really appreciate it. Really appreciate. Hoping for more help from you. Some time I can understood. And its so clear when I know. Thank J ❤

@ivy :

Because, we have 7 x 104 - 6 x 102

We can first do 7 x 104 = 728
And 6 x 102 = 612

728 - 612 = 116


But if you want to save the hassle of doing manual calculation in Olympaid where time is of essence,


Realise that in 7 x 104 - 6 x 102,


7 x 104 is as good as saying 1there are 104 groups of 7.

That is the same as 2 groups of 7 plus 102 groups of 7.

Which equals 7 x 2 + 7 x 102


So this works out to be :


7 x 104 - 6 x 102

= 7 x 2 + 7 x 102 - 6 x 102

Notice now that 7 x 102 - 6 x 102 is just 7 groups of 102 minus 6 groups of 102, leaving 1 group of 102.

So,


= 7 x 2 + 102
= 14 + 102
= 116

AC provided the more direct way to do the calculation, but in Olympiad setting there's little time so calculations will slow you down.

What I'm trying to show here is all the mental shortcuts and tricks to enable a more efficiency and speed.

This oftens requires that the student understands the logic and idea behind these tips and tricks.

For example, the above was demonstrating the distributive law of multiplication.


Olympiad questions require understanding at a higher level

Ya so true. Just joining for some experience. But when trying all these worksheet. I'm like .. fainted. But love learning and answering all these questions. If have someone to teach, will be best but without , it's a experience anyway. Thank J and AC Lim for being there .

7 odd number from 45 to 69 with average of 53
Total of 7 numbers = 7×53 = 371
371-45-69 = 257
When u divide with 5, get around 257÷5= 51.4
So you know that all the numbers should be around 51, so we should start at 51, 49(left of 51),53 (right of 51), 47(left) ,55(right)
So the odd number ...
45,47,49,51,53,57,69
Middle number = 51

Anyway, this question only requires 3 steps once you've understood the logic :

100 + 2 = 102
7 x 2 = 14
102 + 14 = 116

I like the way both of you explaining to me. So details and some so straightforward sometime. And guide me step by step to get my answers. Thank you. Thumb up and blessedness. ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤

I got answer with decimal and I scared. I thought I'm wrong . So I changed concept

Yes. @J can explain very details always. I like it too.

My approach to your other question :

53 - 45 = 8
69 - 53 = 16

So we can think of it as taking away 16 from 69 , and adding 8 to 45 to make them become 53.


16 - 8 = 8

So this means that overall, we still have 8 to be donated to the other remaining middle 5 numbers to make each of them 53.



i.e to get 53 53 53 53 53

Math have so many ways of doing it. And it's always sound so easy after @J and @ AC lim explained. For 2nd question. I'm struck at 2nd part but now I know that I can do it this way thou. Thank you . ⚘⚘⚘⚘⚘⚘⚘⚘

So, how should we redistribute the remaining 5 numbers to get our results?

They are all odd numbers and have to be different ones.


We can take 2 from one 53 and give it to another 53 to get 51 and 55.


We can also take 4 from one 53 and give it to another 53 to get 49 and 57.

We have one last 53 untouched.


Result :

49 51 53 55 57

Lastly, remember that we had to donate 8 to these 5 numbers to make five 53's.


Working backwards, now we must subtract them away.


We cannot go below 45 or have another 45. And we can only subtract 2 or multiples of 2 in order to get odd numbers.

We can only subtract 2 each from 49, 51,53 and 55. If you tried any other combinations of 4 numbers you will get repeat numbers.

Result :


47 49 51 53 57


The middle number is 51

Oh mine. We doing the same way as u told us " donating method " and its works

Alternatively ,


Once you know you need to donate 8 to the remaining 5 numbers to get five 53's,


Working backwards, you can also subtract 8 from them first to get back your original numbers.

Since we want different numbers, we subtract different multiples of 2 from them.


Take away 6 from one of the 53. Take away 2 from another.


Result : 47 51 53 53 53


Lastly, the only way left is to add 4 to one 53 and subtract 4 from another 53. This leaves the last 53 untouched.


Final result : 47 49 51 53 57


Middle number is still 51

Wowwww you are " Super J "