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 Question from Zach: Two stars and one heart weigh 38 units. One star and one heart weigh 27 units what is the weight of each?

Hi Zach,

I'm going to solve a similar problem.

One star and three hearts weigh 18 units. Two stars and one heart weigh 11 units what is the weight of each?

I want to think of weighing the items using a balance rather than a kitchen or bathroom scale.

Image from pixabay.com

I want to start with one star and three hearts on the left pan of the scale and then I need 18 units of weight on the right to balance. Now add another star and three more hearts to the left pan and you then need to add another 18 units of weight to the right side to maintain the balance. So now you have 2 stars and 6 hearts on the left and 36 units of weight on the right. Next remove 2 stars and 1 heart from the left and according to what's given you need to remove 11 units of weight from the right. Now on the left you have no stars and 5 hearts on the left and 38-1=25 units of weight on the right. Hence 5 hearts weigh 25 units of weight so one heart weighs 5 units of weight. Since 1 star and three hearts weigh 18 units and 1 heart weighs 5 units, one star must weigh 3 units.

Let me solve this again with a different method and less writing. Suppose 1 star weighs S units and 1 heart weighs H units. At the beginning the scale looks like

Left side
Right side
1S + 3H
18
2S + 1H
11

Now on the second weighing add 4 stars and 2 hearts to the left and you will then need 22 more units of weight on the right. Hence you now have

Left side
Right side
1S + 3H
18
6S + 3H
33

Finally remove 1 star and 3 hearts from the left side of the second weighing and you are left with

Left side
Right side
5S + 0H
15

And hence 5 stars weighs 15 units so 1 star weighs 3 units.

Now try your problem,
Penny

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