Question

Please help!! I need step by steps thank you!!

1. 8 more than 10 times a number is negative 56

2. For a field trip 5 students rode in cars and the rest filled nine buses. How many students were in each bus if 472 students were on the trip.

3. A mixture containing 6% Boric Acid is to be mixed with 2 quarts of a mixture which is 15% Boric Acid in order to obtain a solution which is 12% Boric Acid. How much of the 6% solution must be used

Answer 1

1. the number is -6.4
2. 52 students were in each bus (but one)
3. 1 quart

Explanation:

1. Let the number be represented by n. Then ten times the number is 10n, and 8 more than that is 10n+8. The problem statement tells us this sum is -56:

10n +8 = -56

10n = -64 . . . . . . . subtract 8 from both sides

n = -6.4 . . . . . . . . . divide both sides by 10

The number is -6.4.

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2. First of all, you can tell there is an issue with this question because the sum of digits of 472 is 13 and the sum of those digits is 4. This tells you that dividing 472 by 9 will give a remainder of 4, not 5. One of the buses will not be filled. (This process of adding digits until you get a single digit is called "casting out nines". The remaining digit is the remainder from division by 9.)

Let b represent the number of students in each bus. Then the problem statement tells us ...

9b + 5 = 472

9b = 467 . . . . . . subtract 5

b = 467/9 = 51 8/9 . . . . almost 52

52 students rode in 8 of the buses; 51 students rode in the last bus.

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3. Let q represent the number of quarts of 6% solution. Then the total amount of Boric Acid in the mix is ...

15%×2 + 6%×q = 12%×(2+q)

Multiplying by 100 to clear the % signs, we have ...

15·2 +6q = 12(2+q)

30 +6q = 24 +12q . . . . . . eliminate parentheses

6 = 6q . . . . . . . . . . . . . . . . subtract 24+6q from both sides

1 = q . . . . . . divide by 6

1 quart of 6% solution must be used.