Balancing

Since one jar balances one mug and one plate (from figure $B$ ), $3$ jars balance $3$ mugs and $3$ plates:

From figure $C$ , replacing the $3$ plates with $2$ bottles will keep the scale balanced:

From figure $A$ , replacing the $3$ jars with $3$ bottles will keep the scale balanced:

Now, remove two bottle from both sides, which will keep the scale balanced because we remove the same number of objects from both sides:

Since a jar balances a bottle, per figure $A$ , a jar also balances $\boxed{3}$ mugs:

(If you're wondering how I made those pictures, I just took the pic from the problem, cut out the relevant parts, and manipulated them in Microsoft Word.)

let $J$ = jug, $B$ = bottle, $M$ = mug, and $P$ = plate

in figure $A$ $\implies$ $J=B$ $\color{#D61F06}(1)$

in figure $B$ $\implies$ $J=M+P$ $\color{#D61F06}(2)$

in figure $C$ $\implies$ $3P=2B$ $\color{#D61F06}(3)$

Algebraic solution:

From $\color{#D61F06}(3)$ , we can say that $P=\dfrac{2}{3}B$ , substituting this in $\color{#D61F06}(2)$ , we obtain, $J=M+\dfrac{2}{3}B$ . But $J=B$ , so $J=M+\dfrac{2}{3}J$ . Simplifying, we get $\dfrac{1}{3}J=M$ .

Finally, $J=3M$

Relevant wiki: Balance Puzzles

Let $J =$ jug, $B =$ bottle, $P =$ plate, and $M =$ mug.

$J = B$ $J = M + P$ $3P = 2B$

$\implies$

$3P = 2J$ $3(J - M) = 2J$

$\implies J = 3M$ .

Here are the three figures written algebraically... j = b j = m + p 3p = 2b

So you want to find x j = X * m

If we look at figure 2 we see that we have mugs and plates equal to 1 jug. So we have to figure out the weight of mugs and plates relative to a jug.

Start by looking at Figure 3, you have 3 plates equal to 2 bottles. Since jugs and bottles are equal you could also say 3 plates = 2 jugs. Solving for plates we know 1 plate = 2/3 jugs

If we substitute that into Figure 2 we can solve for mugs j = m + 2/3 A mug represents 1/3 of a jug.

Therefore it takes 3 of them to be equal.

Jug = bottle

2(jug) = 2(bottle) and 2(bottle) = 3(plate)

~ thus ~ 2(jug) = 3(plate)

And 2(jug) = 2(mug) + 2(plate)

~ thus ~ 2(mug) + 2(plate) = 3(plate)

So let's say mug is a and plate is b

2a + 2b = 3b

2a = 3b - 2b

2a = 1b

~ therefore ~ 2(mug) = 1(plate)

So 1(mug) + 1(plate) = 1(mug) + 2(mug)

And if 1(jug) = 1(mug) + 1(plate)

Then 1(jug) = 1(mug) + 2(mug)

~ therefore ~ 3 mugs will balance the jug

As you can see in the C balance:

2 bottles=3 plates

So, 1 bottle= 1.5 plates

As per the A balance:

1 bottle= 1 jug

So, 1 jug= 1.5 plates

From balance B:

1 jug= 1 mug+1 plate

So, 1 mug = 0.5 plates

Therefore, 1 jug= 3 mugs

By given figure we have,

$J = M + P$

$3P = 2J$

First equation gives,

$3J = 3M + 3P$

$= 3 M + 2J$ by second equation.

$→ J = 3M$

1. From A and C: 2 jugs = 3 plates
2. Hence 1 jug = 1.5 plates
3. From B: 1 jug = 1 plate and 1 cup

Hence from 2 and 3: 1 cup = 0.5 plates 1 jug = 1.5 plates = 3*0.5 plates = 3 cups

Combing A and B, we see that 1 bottle weighs the same as one mug and a plate. Replacing the two bottles on C with 2 mugs and two plates retains the balance on the C scale, So does removing two plates from both sides of C. What's left is one plate on the left and and two mugs on the right. Therefore, one plate weighs the same as two mugs. Returning to B, we then replace the one plate on the right with two mugs, retaining B\s state of equilibrium, proving that one jug weighs the same as three mugs.

Since 1 jug = 1 bottle (A), 2 bottles = 2 jugs, also = 3 plates (C). So 1 jug = 1.5 plates, so 1.5 plates = 1 cup + 1 plate (B). Therefore 1 cup must = .5 plate. So 1jug = 1.5 plates or 3 cups

From A, 1 jug = 1 Bottle

From C, 1 bottle = 1½ plates

Therefore 1 jug = 1½ plates

From B, 1 cup = ½ plate, so that plate can be replaced with 2 more cups to balance, which makes 3.

Quick and easy headcount solution, I'll use j for jug, b for bottle, p for plate and m for mug. From C we get 3p=2b, so if in A 1j=1b, then 2j=2b=3p. From B we get 1j=1m+1p, so 2j=2m+2p, which we previously found out is also 3p. So 2m+2p=3p, so 2m=1p. Since 2j=3p=6m, 1j=3m. Final answer 3 .

1 jug = 1 bottle, 3 plates = 2 bottles, so 3/2 or 1.5 plates = 1 bottle.This means that 1 jug = of a plate, so we can see that 1 jug = 1 plate + 1 mug, so 3/2 of a plate = 1 plate + 1 mug, telling us that 1/2 of a plate = 1 mug, allowing us to easily deduce that 1 jug = 3 mugs.