### Brain Teasers

# Bags, Marbles, and Scales

Math
Math brain teasers require computations to solve.

In front of me, I have level scales. On each side, there are 3 bags. In each bag, there are some marbles. There are 150 Marbles in total. Using the clues below, work out how many marbles are in each bag.

Bag 1 + Bag 2 = 56 marbles

Bag 1 + Bag 3 = 54 marbles

Bag 2 + Bag 3 = 40 marbles

-------------------

Bag 4 + Bag 5 = 61 marbles

Bag 4 + Bag 6 = 37 marbles

Bag 5 + Bag 6 = 52 marbles

Bag 1 + Bag 2 = 56 marbles

Bag 1 + Bag 3 = 54 marbles

Bag 2 + Bag 3 = 40 marbles

-------------------

Bag 4 + Bag 5 = 61 marbles

Bag 4 + Bag 6 = 37 marbles

Bag 5 + Bag 6 = 52 marbles

### Answer

Bag 1 = 35 marblesBag 2 = 21 marbles

Bag 3 = 19 marbles

Bag 4 = 23 marbles

Bag 5 = 38 marbles

Bag 6 = 14 marbles

To work this out, since there are 150 marbles, and the scales are level, both sides of the scales must have 75 marbles. If Bag 1 + Bag 2 = 56 marbles then Bag 3 = (75-56) 19 marbles. So Bag 1 = 35 marbles and Bag 2 = 21 marbles. The same is done with Bags 4,5, and 6. Bag 4 + Bag 5 = 61 marbles. So Bag 6 = (75-61) 14 marbles. So Bag 4 = 23 marbles and Bag 5 = 38 marbles.

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## Comments

I don't believe it, I got it right! chance? Very clever and well done

Good one, although the part about the scales and the 150 total need not even be part of the teaser. If we know there are 2 more marbles in bags 1 and 2 than in bags 1 and 3, then 2 and 3 MUST be 21 and 19 to equal 40. Same for 4,5, and 6.

verrrryyyy eazzzzzeeee!!!

Yes, the part about having 150 marbles and level scales is not necessary. I didn't even use that part at all. Fun, but way easy!

easy but fun.

I didn't understand the idea behind the scale, but I used a system of equations.

A=Bag 1

B=Bag 2

C=Bag 3

1. A + B = 56

2. A + C = 54

3. B + C = 40

Subtracting equation 3 from 1, you get

4. A - C = 16

Adding equation 4 and 2 you get

A + A = 70 = 2A

A = 35.

And that checks with the answer.

A=Bag 1

B=Bag 2

C=Bag 3

1. A + B = 56

2. A + C = 54

3. B + C = 40

Subtracting equation 3 from 1, you get

4. A - C = 16

Adding equation 4 and 2 you get

A + A = 70 = 2A

A = 35.

And that checks with the answer.

I went about this completely wrong, but it was very intriguing. . I like the guy with the algebra system....I was thinking of something else, which is why I didn't get the teaser. I know it had me going, because my head hurts now.

very easy for me, cause im a math wizz . i did algebra using variables. worked pretty well nice job, keep em comin

Great Numerous ways to solve; but half the fun was just that

Very easy. I used the algebra method as well!

Very easy. I used the algebra method as well!

Very easy. I used the algebra method as well!

Very Simple, but fun!

I just used the fact that there were a total of 150 marbles on level scales to check my solution, not to solve the problem. It reminded me of the problem that you posted about the minieral water, sandwich, crisps, and fizzy juice. I used the same method in this as I used in that. Anyway, very fun and easy!!

I just used the fact that there were a total of 150 marbles on level scales to check my solution, not to solve the problem. It reminded me of the problem that you posted about the minieral water, sandwich, crisps, and fizzy juice. I used the same method in this as I used in that. Anyway, very fun and easy!!

Mar 09, 2008

You can solve these systems of equations very readily by placing them into a matrix and transforming it into reduced-row-echelon-form. I typed the matrix into matlab as follows, one column for each bag#, and the last column for number of marbles:

x =

1 1 0 0 0 0 56

1 0 1 0 0 0 54

0 1 1 0 0 0 40

0 0 0 1 1 0 61

0 0 0 1 0 1 37

0 0 0 0 1 1 52

for example, the second row reads:

1 of bag1, 0 of bag2, 1 of bag3, 0 of bag 4, 0 of bag5, 0 of bag6, 40 marbles.

Placing into reduced row echelon form:

rref(x)=

1 0 0 0 0 0 35

0 1 0 0 0 0 21

0 0 1 0 0 0 19

0 0 0 1 0 0 23

0 0 0 0 1 0 38

0 0 0 0 0 1 14

the second row implies that 1 of bag2 has 21 marbles. the result is read from the last column.

Scilab is an free alternative to matlab, and can help you solve many of these brain teases with ease.

The fun part is delving into linear albegra and learning WHY rref works like this =)

x =

1 1 0 0 0 0 56

1 0 1 0 0 0 54

0 1 1 0 0 0 40

0 0 0 1 1 0 61

0 0 0 1 0 1 37

0 0 0 0 1 1 52

for example, the second row reads:

1 of bag1, 0 of bag2, 1 of bag3, 0 of bag 4, 0 of bag5, 0 of bag6, 40 marbles.

Placing into reduced row echelon form:

rref(x)=

1 0 0 0 0 0 35

0 1 0 0 0 0 21

0 0 1 0 0 0 19

0 0 0 1 0 0 23

0 0 0 0 1 0 38

0 0 0 0 0 1 14

the second row implies that 1 of bag2 has 21 marbles. the result is read from the last column.

Scilab is an free alternative to matlab, and can help you solve many of these brain teases with ease.

The fun part is delving into linear albegra and learning WHY rref works like this =)

As several people have pointed out, the information about the scales and the 150 total isn't needed. The rest is very easily determined from the three independent systems of equations.

About 20 seconds in my head--a little too easy.

About 20 seconds in my head--a little too easy.

Wow the way I did this was really a lot more complicated than it should have been, haha. Oh well I still got the answer

I like the comments regarding solving it w/o the given info.

Even w/o the given info, I usually solve these

a+b = x

b+c = y

c+a = z

where x, y, z are constants by adding all 3 equations, finding what a+b+c equals, and subtract each equation from the result to figure out a, b, c.

But using logic based on the difference between 2 of the variables and using the 3rd equation can be faster, as I've noticed from trying it out!

Thanks for the good learning.

Even w/o the given info, I usually solve these

a+b = x

b+c = y

c+a = z

where x, y, z are constants by adding all 3 equations, finding what a+b+c equals, and subtract each equation from the result to figure out a, b, c.

But using logic based on the difference between 2 of the variables and using the 3rd equation can be faster, as I've noticed from trying it out!

Thanks for the good learning.

It's true, the information about the scales is not necessary, but it does get you to the answer quicker without solving simultaneous equations (or putting a matrix into reduced row echelon form).

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