Values of Letters: Find the Value of TABLE

[Natasha1]

Homework Statement

Each letters in the words below represents a number less than 10. The values of the letters in each word are then multiplied together to give the result shown.

Given that:
BAT = 90
LET = 168
BET = 105

What is the value of TABLE?

Homework Equations

Could anyone guide me a little on this exercise? Where to start?
BAT = 90 and BET = 105 vary in only one letter and have a difference of 15. But how do I get the value E or A?

Similarly LET = 168 and BET = 105 vary in one letter and have a difference in 63. But how do I find B or L?

The Attempt at a Solution

 

[SteamKing]

Homework Statement

Each letters in the words below represents a number less than 10. The values of the letters in each word are then multiplied together to give the result shown.

Given that:
BAT = 90
LET = 168
BET = 105

What is the value of TABLE?

Homework Equations

Could anyone guide me a little on this exercise? Where to start?
BAT = 90 and BET = 105 vary in only one letter and have a difference of 15. But how do I get the value E or A?

Similarly LET = 168 and BET = 105 vary in one letter and have a difference in 63. But how do I find B or L?

The Attempt at a Solution

You are told that the letters stand for single digit numbers. Write out the various factors of the numbers 90, 105, and 168, and concentrate on the factors which are less than 10 that can be multiplied together to give these three numbers.

For example, 90 = 2 ⋅ 5 ⋅ 9

 

[SammyS]

Also possible: 90 = 6 ⋅ 5 ⋅ 3

 

[haruspex]

BAT = 90 and BET = 105 vary in only one letter and have a difference of 15. But how do I get the value E or A?

90 and 105 result from multiplying the numbers, so the arithmetic difference is not that interesting. Consider the ratio instead.

 

[Thewindyfan]

You could also use systems of equations twice to get rid of the common digit "T" and "B" to solve for remaining variables then find the remaining variables you'd need and multiply the values together like you'd have to.

It'd take awhile because the algebra could get a bit messy, but if you're confident with your algebra I would try that method :)

 

[Samy_A]

I assume the letters stand for integers less than 10.

In that case, why not start by looking at $\frac{\mathtt{BET}}{\mathtt{LET}}$ (see post 3)? That should give you the values of L and B.

 

[Natasha1]

Thank you everyone, I have understood

 

[SammyS]

Thank you everyone, I have understood

So then, what did you get for TABLE ?

 

[Natasha1]

5040

 

[Natasha1]

I guessed in the end, as I couldn't do it algebraically.

 

[Natasha1]

BET = 5 x 7 x 3

LET = 8 x 7 x 3

BAT = 5 x 6 x 3

T.A.B.L.E = 3 x 6 x 5 x 8 x 7 = 5,040

 

[SammyS]

I guessed in the end, as I couldn't do it algebraically.

Some factoring, especially prime factoring can eliminate any guess work, that is if we know that each letter represents some single digit whole number.

 

[haruspex]

I guessed in the end, as I couldn't do it algebraically.

Had you followed my suggestion in post #4, BAT/BET=90/105=6/7. So A/E=6/7. The values of A and E follow immediately.
Samy made an equivalent suggestion in post #6, using BET and LET.

 

[fresh_42]

I claim there are at least two integer solutions. (for the opening statement hasn't forbidden the second one)

 

[SammyS]

I claim there are at least two integer solutions. (for the opening statement hasn't forbidden the second one)

... and what is that solution ?

 

[fresh_42]

-5040 (B,T,L < 0 < 10)

 

[SammyS]

-5040 (B,T,L < 0 < 10)

Looks OK to me.

Come to think of it, there is no mention of integers either.

 

[fresh_42]

Looks OK to me.

Come to think of it, there is no mention of integers either.

Yes, I thought about it. But I've been lazy. Playing with units was easy. And beside ℚ or ℂ, how about p-adic numbers? Playing with units was easy...
EDIT: Forget the p-adics. You'd probably ask me how to define < 10 there.