SOLUTION: The salt from a bin is to be packed into bags. If I put 10 g of salt into each bag, there will be 7 g short. If I put 9 g of salt into each bag, there will be 4 g extra. Find the m

Question 1159835: The salt from a bin is to be packed into bags. If I put 10 g of salt into each bag, there will be 7 g short. If I put 9 g of salt into each bag, there will be 4 g extra. Find the mass of the salt in the bin.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52858) (Show Source): You can put this solution on YOUR website!
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Let N be the number of grams of the salt in the bin. Consider the number N+7. From the condition, the number N+7 is divisible by 10 and gives the remainder 4+7 = 11 equivalent to 2 when divided by 9. The smallest positive integer N+7 with such properties is 20, i.e. N = 20-7 = 13. The next such integer N+7 is 20+90 = 110, i.e. N = 103. The general formula for N is N = 13 + 90k, where k = 0, 1, 2, 3, . . .

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

The wording of the problem is unusual and, I think, open to interpretation. It is, therefore, a bad problem; if the objective of a math problem is to learn mathematics, the reader should not have to spend time trying to decipher (or guess?) the meaning of the problem.

The other tutor reasonably interpreted "7g short" when packing the salt into 10g bags as meaning the total number of grams was 7 more than a multiple of 10 -- i.e., a number of the form 10n+7.

My interpretation would be that you are 7g SHORT of being able to fill another bag, which means there are only 3g left over after you fill as many bags as you can -- i.e., the total number of grams of salt is a number of the form 10n+3.

Then the conditions of the problem say that the total number of grams of salt is a number with units digit 3 which is 4 more than a multiple of 9.

The smallest possible number of grams with that interpretation is easily seen to be 13.

Of course there are an infinite number of possible total number of grams of salt.

Since you are putting 10g of salt in each bag in one case and 9g in each bag in the other case, you can add any multiple of the LCM of 9 and 10, which is 90, to the first answer we found.

So the number of grams of salt in the bin is 13, plus any multiple of 90.

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