# Letters used in place of numbers

• October 1st 2012, 03:27 AM
KayPee
Letters used in place of numbers
Letters can replace numbers in simple mathematical problems. If E and F represent digits and EF + F = FE, then F equals

A. 3
B. 6
C. 8
D. 9

How best can I resolve this problem such that it will be clear to grade six pupils.

Thanks
• October 1st 2012, 04:02 AM
Plato
Re: Letters used in place of numbers
Quote:

Originally Posted by KayPee
Letters can replace numbers in simple mathematical problems. If E and F represent digits and EF + F = FE, then F equals
A. 3 B. 6 C. 8 D. 9
How best can I resolve this problem such that it will be clear to grade six pupils.

But here goes: $(10E+F)+F=10F+E~.$ so $9E-8F=0$ is it clear now?
• October 1st 2012, 04:17 AM
TwoPlusTwo
Re: Letters used in place of numbers
Edit: Plato beat me to it (plus I was wrong (Giggle))
• October 1st 2012, 04:57 AM
KayPee
Re: Letters used in place of numbers
Quote:

Originally Posted by Plato
But here goes: $(10E+F)+F=10F+E~.$ so $9E-8F=0$ is it clear now?

I followed your approach and my last line was 9E = 8F

why did you equate it to zero i.e. 9E - 8F = 0?
• October 1st 2012, 05:51 AM
emakarov
Re: Letters used in place of numbers
Quote:

Originally Posted by KayPee
I followed your approach and my last line was 9E = 8F

why did you equate it to zero i.e. 9E - 8F = 0?

For all numbers x and y, x = y if and only if x - y = 0.
• October 1st 2012, 07:57 AM
KayPee
Re: Letters used in place of numbers
Quote:

Originally Posted by emakarov
For all numbers x and y, x = y if and only if x - y = 0.

Thanks I got i.e. 9E-8F = 0.

So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?
• October 1st 2012, 08:06 AM
KayPee
Re: Letters used in place of numbers
Quote:

Originally Posted by emakarov
For all numbers x and y, x = y if and only if x - y = 0.

Thanks I got i.e. 9E-8F = 0.

So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?
• October 1st 2012, 08:11 AM
Plato
Re: Letters used in place of numbers
Quote:

Originally Posted by KayPee
Thanks I got i.e. 9E-8F = 0.
So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?

Each of $E~\&~F$ belongs to the set $\{0,1,2,3,4,5,6,7,8,9\}$.
Use simple inspection to determine the solution.
• October 1st 2012, 08:27 AM
KayPee
Re: Letters used in place of numbers
Quote:

Originally Posted by Plato
Each of $E~\&~F$ belongs to the set $\{0,1,2,3,4,5,6,7,8,9\}$.
Use simple inspection to determine the solution.

So from simple inspection E= 8 and F=9.

I think I clear now.Thanks

I hope I can post here if I have any further questions related to the above question.
• October 1st 2012, 10:33 AM
KayPee
Re: Letters used in place of numbers
Quote:

Originally Posted by Plato
Each of $E~\&~F$ belongs to the set $\{0,1,2,3,4,5,6,7,8,9\}$.
Use simple inspection to determine the solution.

Hi Plato

I've got another issue:

I believe the range of possible answers:{0,1,2,3,4,5,6,7,8,9} were selected based on the equation: 9E = 8F.

So if the equation had been something like

1.9R = 9S
As such R and S=1

The possible range must be: {0,1,2,3,4,5,6,7,8,9}

2.For 10x = 8y

The range of possible answers will be: {0,1,2,3,4,5,6,7,8,9,10}

I hope I'm right?
• October 1st 2012, 11:08 AM
Plato
Re: Letters used in place of numbers
Quote:

Originally Posted by KayPee
2.For 10x = 8y
The range of possible answers will be: {0,1,2,3,4,5,6,7,8,9,10}
I hope I'm right?

$10x = 8y$ has no solution.
$x~\&~y$ are digits. No three digit number equals a two digit number.
• October 1st 2012, 11:41 AM
emakarov
Re: Letters used in place of numbers
Quote:

Originally Posted by KayPee
I believe the range of possible answers:{0,1,2,3,4,5,6,7,8,9} were selected based on the equation: 9E = 8F.

The range of E and F was not selected based on the equation 9E = 8F. The problem statement says, "E and F represent digits." Therefore, $0\le E,F\le 9$. In fact, since the problem statement uses two-digit numbers FE and EF, neither E nor F can be 0. Further, 9E = 8F implies that 8F is divisible by 9. But since the GCD(8, 9) = 1, (a generalization of) Euclid's lemma says that F must be divisible by 9. Since F is an integer between 1 and 9, F must be 9. It is also possible to go through all possible variants for F and see for which one 8F is divisible by 9.