<p>Here are three math problems that I am having problems with. Thanks for the help. </p>
<li>Quadrilateral ABCD has a perimeter of 26 and sides of integer lengths. If AB=m, and BC=CD=DA=n, then what is the difference between the greatest and least possible values of n?
A. 3
B. 4
C. 5
D. 6
E. 7</li>
</ol>
<p>What is the best way to solve this problem? Is there a way of doing this algebriaclly, because the explanation for this problem stated that some of the quadrialaterals are impossible to create. They said to try to draw them. Such as a 1, 1, 1, 23. Is there a rule for figuring out if a quadrileral is possible to construct. </p>
<li><p>A certain clock has a minute hand that is exactly twice as long as its hour hand. If point A is at the top of the minute hand, and point B is at the tip of the hour hand, what is the ratio of the distance that point A travels in 3 hours to the distance that point B travels in 9 hours?
A. 1:8
B. 1:4
C. 1:2
D. 2:3
E. 8:1</p></li>
<li><p>The integer 90 can be expressed as the sum ofz consecutive integers. The value of z could be any of the following EXCEPT
A. 3
B. 4
C. 5
D. 6
E. 9</p></li>
</ol>
<p>Is there a mathematical way of solving this problem. The explanation for this problem said to plug in numbers. But that would take too much time on the SAT. Is there a shorter way of doing this problem?</p>
<p>Q1. You need 3n + m = 26, for starters. Also, for a geometrically feasible quadrilateral, you need 3n > m (think of the n + n + n sides lying <em>almost</em> in a straight line; the length of the fourth side will be slightly shorter than the length you would travel, if you went along the equal sides of the quadrilateral (B -> C -> D -> A) to travel from B to A.</p>
<p>If you try for m=3n to get the smallest n value, then m + 3n = 6n = 26
from which n = 26/6 = 3.33 . This is the smallest possible n value, but n must also be integer; so min_n = ceiling(3.33) = 4</p>
<p>For max<em>n, try setting m=1 (smallest integer); then 1 +3n = 26, or n=8.33 . This is the largest possible n value, but n must also be integer; so max</em>n = floor( 8.33) = 8 .</p>
<p>Max<em>n - Min</em>n = 8 - 4 = 4 (unless I screwed up the numbers, which at my age is entirely possible).</p>
<p>Q3. If you add six consecutive integers, you will be adding either
even + odd + even + odd +even + odd = (3 evens) + (3 odds) = even + odd = odd number, <> even number like 90
OR
odd +even + odd +even + odd +even = (3 evens) + (3 odds) = even + odd = odd number, <> even number like 90
So, adding six consecutive integers can never give you an even number (e.g. 90).</p>
<p>Q2. Say the hour hand has a length of “r”. Then, in 12 hours, it would travel one full revolution, with a circumference of 2(pi)r. So in 9 hours, it’s gone 3/4 of that distance, so its moved 3(pi)r/2.</p>
<p>The minute hand is twice as long, so its radius is “2r”. In one hours, it moves 2(pi)*(2r) = 4(pi)r. In three hours, it’s moved 12(pi)r. So the ratio of the distance traveled by A vs B is 3/2:12 = 6:24 = 1:4</p>
<p>Q1. Not quite, optimizer. 3n must be > m, but 3n+m=26, so 3n must be more than half of 26. The smallest n for which this is true is 5. The biggest possible n is, as you point out, 8. (It can’t be 9 beacuse 3x9>26.) This leaves 2 for m and if you picture the quadrilateral, it’s a traepzoid. So the difference between max(n) and min(n) is 8-5=3.</p>
<p>As a check, if n =4, m would be 14, so 3n would be <m.</p>
<p>Optimizer - Q1:
I think you did make a slight math error (26/6 = 4.33, not 3.33, so that the min = 5, not 4 and the difference is 3).
TanMan - Q2:
The question is asking for the ratio of A to B is it not? You seem to have provided the ratio of B to A. Should the answer not be 4:1, not 1:4? A is the minute hand and it is longer and travels around the clock face 3 times. Hence the answer has to be a number greater than 1 since it is being compared to the hour hand which is shorter and travels around less than one revolution. 3 revolutions compared to 3/4 of a revolution = 4 to 1, right?</p>
<p>Tanman - Q2:
Actually, it looks like you may have made a math error on this question also. In addition to reversing the ratio, should the division also not yield an 8:1 ratio, not 4 to 1 (using your numbers that resulted in 12 and (3/2)?</p>
<p>when I did this question i recieved an answer of 8:1, but the book said that the answer was 2:3. I found other errors in that same practice test and I think this is another error. The explanation did not even make sense so it did not help. Anyone else think that the answer is 8:1?</p>