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Another Algebra Problem

A student got 50% of the questions on an algebra test correct. If he answered 10 out of the first 12 questions asked correctly but missed 3/4 of the remaining questions, how many questions were on the test?

So algebraically we need to setup an equation here. The sum of the 10 correctly answered questions and the remaining questions that were answered correctly divided by the total number of questions is 0.5. Let y represent the number of remaining questions answered correctly. Let x represent the number of questions in total. Then;

10/x + y/x = 0.5
10 + y = 0.5x equation 1

However we were told that 3/4 of the remaining question were answered wrong. Therefore 1/4 was of the remaining questions were answered correctly.

Thus,
y= 1/4(x-12)
y= 0.25(x-12) equation 2

Substitute equation 2 in equation 1 we have;

10 + 0.25(x - 12) = 0.5x
10 + 0.25x -3 = 0.5x
7= 0.25x
x= 28

Therefore there were a total of 28 questions on test. In first section 10 of 12 were answered correctly. This means that there were 16 questions in second section and only 4 answered correctly.

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YET ANOTHER ALGEBRA I CONCEPT QUESTION!
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A family makes a 43.25 km trip in 5.5 hours. On the first part of the trip they crossed a lake in a canoe paddling at 12 km/h. For the rest of the trip, they hiked on a scenic trail. If their average walking speed was 5 km/h, how far did they walk?

Let t represent the time traveled by walking. Since speed (v) is equal to distance (d) divided by time (t). That is (v=d/t), then time is distance divided by speed(t=d/v).

Therefore the total time taken for journey can be represented by;

(43.25-t)/12 + t/5 = 5.5

To get rid of fraction we multiply by the LCM of 60.

60*(43.25-t)/12 + 60*t/5 = 60* 5.5
5*(43.25-t) + 12t = 330
216.25 - 5t + 12t = 330
7t = 330 - 216.25
t = 113.75/7
t = 16.25 km

Thus the family walked 16.25 kilometers.

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