A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures inches high and inches wide. What is the area of the border, in square inches?
Solution
In order to find the area of the frame, we need to subtract the area of the photograph from the area of the photograph and the frame together. The area of the photograph is square inches. The height of the whole frame (including the photograph) would be , and the width of the whole frame, . Therefore, the area of the whole figure would be square inches. Subtracting the area of the photograph from the area of both the frame and photograph, we find the answer to be .
==Video Solution==
https://youtu.be/P89ysBIaPfs ~savannahsolver
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker . What is the missing digit of this -digit number?
Solution
Since all the eleven members paid the same amount, that means that the total must be divisible by . We can do some trial-and-error to get , so our answer is
~SparklyFlowers
In the given figure, hexagon ABCDEF is equiangular, ABJI and FEHG are squares with areas 18 and 32 respectively, triangle JBK is equilateral and FE=BC. What is the area of triangle KBC?
[figure]
Solution
Square ABJI's side length is sqrt{18}=3sqrt{2}, and since Triangle JBK is equilateral, that means that the side length of Triangle JBK would be 3sqrt{2}, which means that KB, the height of Triangle KBC would be 3sqrt{2}. Square FEHG's side length would then be sqrt{32}=4sqrt{2}, and because line FE = line BC, The base of Triangle KBC would be 4sqrt{2}. The formula to finding the area of a triangle is the frac{Height cdot Base}{2}, which means that Triangle KBC's area would just be frac{3sqrt{2}cdot4sqrt{2}}{2}, and the answer is boxed{12}.
~MMA2024
==Video Solution==
https://youtu.be/mUBhWTlvuLQ
~savannahsolver
A (frac{1}{4}) fraction of the balls in a bag are blue. (frac{1}{3}) of the non-blue balls are red. The rest are green. If there are 6 green balls, how many balls are in the bag?
Solution
Blue = 1/4 of total. Non-blue = 3/4. Red = 1/3 of 3/4 = 1/4 of total. Green = 3/4 - 1/4 = 1/2 of total. So 1/2 × total = 6 → total = 12. Hmm that gives 12. Official answer is D:24.
What is the correct ordering of the three numbers frac{5}{19} , frac{7}{21} , and frac{9}{23} , in increasing order?
Solution
The value of frac{7}{21} is frac{1}{3} . Now we give all the fractions a common denominator.
frac{5}{19} implies frac{345}{1311}
frac{1}{3} implies frac{437}{1311}
frac{9}{23} implies frac{513}{1311}
Ordering the fractions from least to greatest, we find that they are in the order listed. Therefore, our final answer is boxed{textbf{(B)} frac{5}{19}<frac{7}{21}<frac{9}{23}} .