Question

Solve each of the word problems. a. Ten less than four times a number is 14. What is the number? b. The smaller of two integers, x and x + 3, is multiplied by 3, and then added to the other integer. The sum of the two integers is 100, what is the larger integer? c. The sum of three consecutive numbers is 129. What is the difference between the largest and the smallest number? d. How many liters of water must be added to 50 liters of 70% ethyl alcohol to reduce it to a mixture of 40% ethyl alcohol concentration? The Nature of Mathematics 25

Answer 1

a) 6.

b) 24.25 (but this is not an integer, so there is no solution).

c) The difference is 2 and the smallest number is 42.

d) We should add 37.5 liters of water.

Explanation:

a) Ten less than four times a number is 14.

If the number is x, then four times the number would be 4x and ten less than it would be 4x-10.
`4x-10=14\\4x=24\\x=24/4\\x=6`

The number is 6.

b) The small integer is x and the larger integer is x+3, if we multiply the smaller integer by 3, we get 3x. Now we will sum 3x to x+3 and the result should be 100.

`3x+x+3=100\\4x=97\\x=97/4\\x=24.25`

This problem has no solution because x=24.25 but the problem says x is supposed to be an integer.

c) If we name the smaller number x, then the second one would be x+1 and the third one would be x+2.

`x+(x+1)+(x+2)=129\\3x+3=129\\3x=126\\x=126/3\\x=42`

The smaller number is 42 and the difference between the smallest and the largest is 2 (we didn't have to solve the equation to tell this, since they are 3 consecutive numbers, the first one is x and the last one is x+2, giving us a difference of 2)

d) Let's call x the number of liters of water we should add. Now we have that our current mixture has 50 liters of 70% ethyl alcohol, therefore it has 30% of water. Once we have added the x liters we will have 50 + x liters and we want this mixture to be 60% of water
`.30(50L)+x=.60(50L+x)\\15L+x=30L+.60x\\x-.60x=30L-15L\\.40x=15L\\x=15L/.40\\x=37.5L`.

We should add 37.5 L of water.