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WEIGHTY PROBLEMS !

1. Bags and Beads

you are given 5 bags. There are 10 beads in each of the bags. In four of the bags, the beads each weigh 10 kilograms. In the remaining bag, each bead weighs only 9 kilograms. All the bags and beads look identical. You must find out which bag has the lighter beads. The problem is that all the bags look identical and all the beads look identical. You can use a scale, but it has to be a single-tray scale, not a two-tray balance scale. Also, you may use the scale only once. How can you find out which bag has the lighter beads?

A - Label the bags from 1 to 5. Take 1 bead out of Bag 1, and label it 1. Take 2 beads out of Bag 2, and label them both with a 2. Take 3 beads out of Bag 3, and label each with a 3. Continue this pattern with Bags 4 and 5. Put these 15 beads on the tray of the scale.
If all 15 weighed 10 kilograms, the scale would register 150 kilograms. But since one or more of the beads weighs only 9 kilograms, the scale will register less than 150. Subtract the number on the scale from 150. Your answer will tell you the number of the bag with the lighter beads. (If the scale registers 148, it’s bag #2. If the scale registers 145, it’s bag #5.)

2. GOLD COINS

Now you should be able to solve this variation, no problem:

You have 10 bags of gold coins, 10 coins per bag, 10 grams per coin, but one bag of coins weigh only 9 grams per coin (because of low quality). How do you find out which bag contains low quality gold coins? You may use a scale only one time.

Answer

step 1: we name all the bag of gold coins as #1, #2, #3......#8, #9, and #10

step 2: we put 1 coin from bag #1, 2 coins from bag #2, 3 coins from bag #3.........8 coins from bag #8, 9 coins from bag #9, and 10 coins from bag 10 onto the scale. Find out the total weight.

step 3: the total weight should have been 10 grams X

(1+2+3+4+5+6+7+8+9+10=55) = 550 grams if all coins are the same (10grams each).

step 4: Subtract the total of step 2 from total of step 3.

Conclusion: If step 4 results 1 gram, then bag #1 is the low quality coins, if step 4 results 2 grams, then bag #2 is the one, if step 4 results 3 grams, then bag #3 is the one.......etc.

3. NINE MARBLES

You have 9 marbles: 8 of them weigh 1 ounce each; 1 weighs 1.1 ounce. The 9 marbles are all uniform in size, appearance and shape. You have access to a balance scale containing 2 trays - you may use the balance 2 times. You must determine which of the 9 marbles is the heavier one using the balance only 2 times.

Answer

Place 3 marbles on each tray.
- If the first weighing doesn't balance, remove all the marbles from the lighter side, and place one marble on each tray from the heavier tray. The heavier side is the 1.1 ounce marble, but if they balance, then the marble from the heavier tray from the first weighing that was not weighed in the second weighing is the heavier one(1.1).
- If the marbles balance on the first weighing, remove the marbles from the trays, and place 2 of the remaining unweighed marbles on the trays, one on each tray. If one is heavier, it is the heavier marble(1.1), but if they balance, the remaining unweighed marble is the heavier one.

Now thry these without the answers

4. NINE GOLD COINS

This one is a little more difficult because you are not told if the object is heavier or lighter.

You have 9 gold coins. All 9 coins look exactly the same but one coin is a fake and is either lighter or heavier than the other 8 coins. You have a scale - balance type with 2 trays - but can only load it twice. How do you find the fake gold coin?

5. CASE OF THE COUNTERFEIT COINS

You have 12 identical-looking coins, one of which is counterfeit. The counterfeit coin is either heavier or lighter than the rest. The only scale you have to use is a simple balance. Using the scale only three times (Note: not loading, but using for balancing), find the counterfeit coin.