So the obvious guess is 1999 or 2006.

So the answer is 1999 or 2006.In summary, The conversation is about a geometry problem that involves finding A, which is equal to (4/5)(18292 + 12982). The speaker had trouble solving this problem in under 3 minutes and was looking for an algebraic manipulation to make it easier. Another person suggested factoring and using shortcuts to solve it quickly. The speaker also mentioned using modular arithmetic to narrow down the possible values of A, which turned out to be 1999 or 2006.
  • #1
ephedyn
170
1

Homework Statement



Find A:

A2 = (4/5)(18292 + 12982)

in about 3min, because this comes at the end of a rather difficult geometry problem with 6 min for the entire question. (edit: Yes, calculators weren't allowed because it was a competition. I have verified that everyone comes to this step, so I didn't make any mistakes before that.)

Homework Equations



None.

The Attempt at a Solution



I'm quite sure there's some algebraic manipulation to help you with this one, but I can't think of it. I did the 18292 and 12982 the 'brute force' way, then added them together for the sum. This step took the longest - more than 3min for me.

Then I converted this to standard form so as to mentally estimate the common logarithm to 3 decimal points. I also know common lg 4 and lg 5 by memory to 3 decimal points - then I took a 'guess' at A:

lg A = (lg 4 + lg(18292+12982) - lg 5)/2

Comes around to 2 * 103~ rather quickly, correct a bit and 'try' a few values, and I found 2006 eventually (edit: , which is the correct answer.)

Also did think of putting it in (1829+1298)2 - 2*1829*1298, but it still takes about as long. Alternatively, I had a suggestion to use 18002 + 292 + 2*1800*29, and similar for 12002 + 982 + 2*1200*98, which takes about as long for me.

Does anyone see an algebraic manipulation to make this easier? Thanks in advance ;)
 
Last edited:
Physics news on Phys.org
  • #2
Not really algebraic manipulation -- just factoring.
A2 = 4/5[18292 + 12982]
= 4/5[592 * 312 + 542 * 112 * 22]
= 4/5[592(312 + 112 * 4)]
= 4/5[592(961 + 484)]
= 4/5[592(1445)] = 4/5[592 * (5 * 289)]
= 4[592 * 172]

==> A = 2*59*17 = 2006
 
  • #3
Oh, I didn't see 59^2 and 17^2 (don't come across these primes often)! Thanks very much. I wish you a good weekend.
 
  • #4
Would 1829=1830-1 be useful?
 
  • #5
What was the problem? Working it differently might have lead to something with simpler arithmetic at the end.


If you're really serious about speed arithmetic, then you might want to practice multiplication; you should be able to compute (4/5)(1829^2 + 1298^2) in under three minutes the direct way... and even faster with shortcuts (e.g. computing 1830^2 and 1300^2 first, and adjusting to get the correct answer)


Also, I assume you're practicing from three-year old problems? It is common to have problems involve the year somehow -- so if you know there's an answer around 2000, the very first thing you should guess is the year. :smile: Maybe even memorize other interesting things -- such as its prime factorization.


You can use modular arithmetic to pick the answer out of a narrow range. e.g.

A^2 = (4/5)(1829^2 + 1298^2) = (4*2) (2^2 + 2^2) = 8*8 = 1 (mod 9)
A = 1 or 8 (mod 9)

A^2 = (4/5)(1829^2 + 1298^2) = (4*3) (2^2 + 3^2) = 5 * 6 = 2 (mod 7)
A = 3 or 4 (mod 7)

Based on the modulo 9 criterion, the possibilties nearest 2000 are
..., 1990, 1997, 1999, 2006, 2008, ...

And modulo 7, they are
..., 1991, 1992, 1998, 1999, 2005, 2006, ...
 

1. What is mental calculation of squares?

Mental calculation of squares is the process of finding the square of a number without using a calculator or any other external aid. It involves using mental strategies and techniques to quickly and accurately calculate the square of a number.

2. Why is mental calculation of squares important?

Mental calculation of squares is an important skill to have because it improves your mental math abilities and helps you solve problems quickly and efficiently. It also allows you to check your work when using a calculator and enables you to estimate answers for real-life situations.

3. What are some strategies for mentally calculating squares?

Some strategies for mentally calculating squares include using the square of a nearby number, breaking down the number into smaller factors, and using the properties of squares. Other techniques may involve using visualizations, patterns, and memorization of commonly used squares.

4. Can anyone learn how to mentally calculate squares?

Yes, anyone can learn how to mentally calculate squares with practice and the right strategies. It may take some time and effort, but with consistent practice, anyone can improve their mental math skills, including calculating squares.

5. How can I improve my mental calculation of squares?

To improve your mental calculation of squares, you can practice regularly, use different strategies, and challenge yourself with different numbers and types of problems. You can also seek help from a tutor or use online resources and apps that provide practice and guidance for mental math skills.

Similar threads

Replies
4
Views
978
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
116
  • Engineering and Comp Sci Homework Help
Replies
2
Views
2K
  • Biology and Medical
9
Replies
287
Views
18K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Quantum Physics
3
Replies
87
Views
5K
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Precalculus Mathematics Homework Help
Replies
13
Views
3K
Back
Top