Math Question

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1

<p>If y is directly proportional to x, which of the following could express y in terms of x?</p>

<p>a) 4x (right answer)
b) x+4
c) x^2
d) 4/x
e) (x+1)/4</p>

<p>The average age of the people in a certain group was 16 years before one of the members left the group and was replaced by someone who is 12 years older than the person who left. If the average age of the group is now 18 years old, how many people are in the group?</p>

<p>I got the answer right. But I want to see if I did it in a correct, logical method</p>

2

<p>The first question is just to know the definition of direct proportionality.
"Two quantities x and y are said to be directly proportional, proportional, or “in direct proportion” if y is given by a constant multiple of x , i.e., y=cx for a constant c. "</p>

<p>Second question:</p>

<p>The sum of the ages for the first group is 16x, where x represents the number of people in the group. The sum of the ages for the second group is 18x.</p>

<p>If the second group replaces the one person with another person 12 years older, then the sum of ages for the second group is 12 greater than the sum of the first group:</p>

<p>16x + 12 = 18x</p>

<p>then solve. 2x = 12; x = 6</p>

3

<p>I thought directly proportional was yk=x?</p>

4

<p>yk = x and y = kx are the same. The first equation is equivalent to y = (1/k)x, which is another constant (provided k is not zero).</p>

5

<p>Then if it is y=kx…</p>

<p>couldn’t it also equal (x+1)/4?</p>

6

<p>(x + 1)/4 = (1/4)x + 1/4.</p>

<p>This does not have the form y=kx where k is a constant. It has the form y=kx+b.</p>

<p>The graph of an equation which represents a direct proportion is a line through the origin.</p>

<p>The y-intercept of the graph of this equation is (0,1/4).</p>

7

<p>A little lesson on Direct Variation:</p>

<p>The following are all equivalent ways of saying the same thing:
(1) y varies directly as x<br>
(2) y is directly proportional to x
(3) y=kx for some constant k
(4) y/x is constant
(5) the graph of y=f(x) is a nonvertical line through the origin.</p>

<p>For example, in the equation y=5x, y varies directly as x. Here is a partial table of values for this equation.</p>

<p>x 1. 2… 3… 4
y 5. 10. 15. 20</p>

<p>Note that we can tell that this table represents a direct relationship between x and y because 5/1=10/2=15/3=20/4. Here the constant of variation is 5. </p>

<p>The graph of the equation is a line through the origin with a slope of 5.</p>

<p>Note that we can tell that the graph represents a direct relationship between x and y because it is a nonvertical line through the origin. The constant of variation is the slope of the line, in this case m=5.</p>

8

<p>Here is a simple example with several different solutons:</p>

<p>If y varies directly as x and y=5 when x=8, then what is y when x=24?</p>

<p>Solution 1: Since y varies directly as x, y = kx for some constant k. We are given that y = 5 when x = 8, so that 5 = k(8), or k = 5/8. Therefore y = 5x/8. When x = 24, we have y = 5(24)/8 = 15.</p>

<p>Solution 2: Since y varies directly as x, y/x is a constant. So we get the following ratio: 5/8 = y/24. Cross multiplying gives 120 = 8y, so that y = 15.</p>

<p>Solution 3: The graph of y = f(x) is a line passing through the points (0, 0) and (8, 5). The slope of this line is (5 – 0)/(8 – 0) = 5/8. Writing the equation of the line in slope-intercept form we have y = 5/8 x. As in solution 1, when x = 24, we have y = 5(24)/8 = 15.</p>

<ul>
<li>Solution 4: To get from x = 8 to x = 24 we multiply x by 3. So we have to also multiply y by 3. We get 3(5) = 15.</li>
</ul>

<p>Note that Solution 4 is the quickest!</p>