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The area of polygon , in square units, is
The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?
A ream of paper containing sheets is cm thick. Approximately how many sheets of this type of paper would there be in a stack cm high?
A "stair-step" figure is made of alternating black and white squares in each row. Rows through are shown. All rows begin and end with a white square. The number of black squares in the row is
If , the largest number in the set is
The product of the 9 factors
The fraction halfway between and (on the number line) is
A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled is
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are , and . The area of the square is
If you walk for minutes at a rate of and then run for minutes at a rate of how many miles will you have gone at the end of one hour and minutes?
The difference between a sales tax and a sales tax on an item priced at before tax is
How many whole numbers between and contain the digit ?
The ratio of boys to girls in Mr. Brown's math class is . If there are students in the class, how many more girls than boys are in the class?
If your average score on your first six mathematics tests was and your average score on your first seven mathematics tests was , then your score on the seventh test was
Nine copies of a certain pamphlet cost less than while ten copies of the same pamphlet (at the same price) cost more than . How much does one copy of this pamphlet cost?
If the length and width of a rectangle are each increased by , then the perimeter of the rectangle is increased by
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January fall that year?
Mr.Green receives a raise every year. His salary after four such raises has gone up by what percent?
Assume every 7-digit whole number is a possible telephone number except those that begin with or . What fraction of telephone numbers begin with and end with ?
Note: All telephone numbers are 7-digit whole numbers.
King Middle School has students. Each pupil takes classes a day. Each teacher teaches classes. Each class has students and teacher. How many teachers are there at King Middle School?
In a magic triangle, each of the six whole numbers is placed in one of the circles so that the sum, , of the three numbers on each side of the triangle is the same. The largest possible value for is
Five cards are lying on a table as shown.
Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?
1.We could go at it by just multiplying it out, dividing, etc, but there is a much more simple method.
Noticing that multiplying and dividing by the same number is the equivalent of multiplying (or dividing) by , we can rearrange the numbers in the numerator and the denominator (commutative property of multiplication) so that it looks like
Notice that each number is still there, and nothing has been changed - other than the order.
Finally, since all of the fractions are equal to one, we have , which is equal to .
Thus, is the answer.
However, if you want to multiply it out, then it would be.
That would be, which is 1. Therefore, the answer is
2.
One possibility is to simply add them. However, this can be time-consuming, and there are other ways to solve this problem. We find a simpler problem in this problem, and simplify ->
We know , that's easy - . So how do we find ?
We rearrange the numbers to make . You might have noticed that each of the terms we put next to each other add up to 10, which makes for easy adding. . Adding that on to 900 makes 945.
945 is
Instead of breaking the sum and then rearranging, we can start by rearranging:
We can use the formula for finite arithmetic sequences.
It is () where is the number of terms in the sequence, is the first term and is the last term.
Applying it here:
3.We immediately see some canceling. We see powers of ten on the top and on the bottom of the fraction, and we make quick work of this:
We know that , so
So the answer is
4.
Obviously, there are no formulas to find the area of such a messed up shape, but we do recognize some shapes we do know how to find the area of.
If we continue segment until it reaches the right side at , we create two rectangles - one on the top and one on the bottom.
We know how to find the area of a rectangle, and we're given the sides! We can easily find that the area of is . For the rectangle on the bottom, we do know the length of one of its sides, but we don't know the other.
Note that , and , so we must have
The area of the bottom rectangle is then
Finally, we just add the areas of the rectangles together to get .
Let be the area of polygon . Also, let be the intersection of and when both are extended.
Clearly,
Since and , .
To compute the area of , note that
We know that , , , and , so
Thus
Finally, we have
This is answer choice
5.To get the fraction, we need to find the number of people who got grades that are "satisfactory" over the total number of people.
Finding the number of people who got acceptable grades is pretty easy. 5 people got A's, 4 people got B's, 3 people got C's and 3 people got D's. Adding this up, we just have .
So we know the top of the fraction is 15. Only 5 people got "unacceptable" scores, so there are scores.
is our fraction, so is the answer.
6.
We could solve the first equation for the thickness of one sheet of paper, and divide into the 2nd equation (which is one way to do the problem), but there are other ways, too.
Let's say that . So by multiplying by this fraction, we SHOULD get the number of sheets in 7.5 cm. Solving gets
is
We can set up a direct proportion relating the amount of sheets to the thickness because according to the problem, all the papers have the same thickness. Our proportion iswhere is the number we are looking for. Next, we cross-multiply to get so which is
7.
The best way to solve this problem is to find patterns and to utilize them to our advantage. For example, we can't really do anything without knowing how many squares there are in the 37th row. But who wants to continue the diagram for 37 rows? And what if the problem said 100,000th row? It'll still be possible - but not if your method is to continue the diagram...
So hopefully there's a pattern. We find a pattern by noticing what is changing from row 1 to row 2. Basically, for the next row, we are just adding 2 squares (1 on each side) to the number of squares we had in the previous row. So each time we're adding 2. So how can we find , if is the number of squares in the row of this diagram? We can't just say that , because it doesn't work for the first row. But since 1 is the first term, we have to EXCLUDE the first term, meaning that we must subtract 1 from a. Thus, . So in the 37th row we will have .
You may now be thinking - aha, we're finished. But we're only half finished. We still need to find how many black squares there are in these 73 squares. Well let's see - they alternate white-black-white-black... but we can't divide by two - there aren't exactly as many white squares as black squares... there's always 1 more white square... aha! If we subtract 1 from the number of squares (1 white square), we will have exactly 2 times the number of black squares.
Thus, the number of black squares is
36 is
Note that each row adds one black and one white square to the end of the previous one, and then shifts the new row over a little. Since the first row had no black squares, the number of black squares in any row is one less than the row number.
Now that this has been established, we just have
8.Since all the numbers are small, we can just evaluate the set to be
The largest number is , which corresponds to .
9.First doing the subtraction, we get
We notice a lot of terms cancel. In fact, every term in the numerator except for the and every term in the denominator except for the will cancel, so the answer is , or
If you don't believe this, then rearrange the factors in the denominator to get
Everything except for the first term is , so the product is
10.The fraction halfway between and is simply their average, which is
Set aside the a. Take the average of b and c. For this example, the average of 3 and 5 is 4. Then multiply those numbers. 3*5=15. The number that is exactly in between and will be .
In this example, it would be , which simplifies to .
Therefore, the answer is
11.To find the face opposite , we can find the faces sharing an edge with , so the only face remaining will be the opposite face.
Clearly, and share an edge with . Also, the faces , , and share a common vertex, therefore shares an edge with . Similarly, the faces , , and share a common vertex, so shares an edge with .
The only face doesn't share an edge with is , which is choice
12.We are given the three side lengths of the triangle, so we can compute the perimeter of the triangle to be . The square has the same perimeter as the triangle, so its side length is . Finally, the area of the square is , which is choice
13. minutes is of an hour, so the walking contributes .
Similarly, minutes is of an hour, so the running adds .
Their total is , which is
14.The most straightforward method would be to calculate both prices, and subtract. But there's a better method...
Before we start, it's always good to convert the word problems into expressions, we can solve.
So we know that the price of the object after a increase will be , and the price of it after a increase will be . And what we're trying to find is , and if you have at least a little experience in the field of algebra, you'll notice that both of the items have a common factor, , and we can factor the expression into
is choice
15.
This is a very common type of counting problem that you'll see quite often. Doing this the simple way would take too long, and might even make lots of mistakes.
If you ever learned about complementary counting, this would be the best time to utilize it. Instead of counting how many DO have 2's, why don't we count how many that DON'T?
So let's find the number of numbers. Obviously, we'd start by subtracting 100 from 400, getting us 300, but we're not done. Since just subtracting includes the number 400, we must subtract one (because 400 isn't allowed - it says between), getting us 299.
So how many numbers are there that DON'T have a 2? Well, we have 2 possibilities for the hundreds digit (1, 3, note that 2 is not allowed), 9 possibilities for the tens digit (1, 3, 4, 5, ... , 9, 0), and 9 possibilities for the ones digit. . However, one of the numbers we counted is , which isn't allowed, so there are numbers without a 2.
Since there are 299 numbers in total and 161 that DON'T have any 2's, numbers WILL have at least one two.
As in the previous solution, get rid of the to make things easier. This gives us the numbers from to .
Let be the event that the first digit is , be the event that the second digit is , and be the event that the third digit is . Then, PIE says that our answer will be .
We have that is just , , and .
Next, is just having something in the form , so there are ways. Similarly, means that we have , so there are again ways. Finally, means that our number is like , so there are ways.
Finally, counts the number of three digit numbers with all three digits , which there is only of: .
Putting this together, our answer will be .
16.Let the number of boys be . It follows that the number of girls is . These two values add up to students, so
The difference between the number of girls and the number of boys is , which is , so the answer is .
17.
If the average score of the first six is , then the sum of those six scores is .
The average score of the first seven is , so the sum of the seven is
Taking the difference leaves us with just the seventh score, which is , so the answer is
Let's remove the condition that the average of the first seven tests is , and say the 7th test score was a . Then, the average of the first seven tests would be
If we increase the seventh test score by , the average will increase by . We need the average to increase by , so the seventh test score is more than , which is clearly . This is choice
18.Let be the cost of the first pamphlet, in dollars. The first part tells us
The second part tells us that
Combining these two parts, the only possible value for is , since represents a monetary value.
Also, just by quickly going over the answers, it is seen that all of the answers satisfy the first statement: The cost of 9 pamphlet is less that 10 dollars.
The second statement states that 10 pamphlets cost more than 11 dollars. Therefore, since there are only two statements and the first one can be satisfied with all of the answer choices, the answer choice that is the most expensive of them all will be the right answer choice, thus the answer is
19.
Let the width be and the length be . Then, the original perimeter is .
After the increase, the new width and new length are and , so the new perimeter is .
Therefore, the percent change is
Assume WLOG that the rectangle is a square with length and width . Thus, the square has a perimeter of . Increasing the length and width by will increase the dimensions of the square to 11x11. Thus, the new square's perimeter is , and because is of , our answer is .
20.January has four full weeks and then three extra consecutive days. Each full week contributes one Tuesday and one Saturday, so the three extra days do not contain a Tuesday and Saturday. Therefore, those three days are Wednesday, Thursday, and Friday.
Wednesday is the 29th day of January, therefore the 22nd, 15th, 8th, and 1st of January are all Wednesdays, so the answer is
21.Assume his salary is dollars. Then in the next year, he would have dollars, and in the next he would have dollars. The next year he would have dollars and in the final year, he would have . As the total increase is greater than , the answer is
Note that we could have generalized this for any integer .
22.An equivalent problem is finding the probability that a randomly selected telephone number begins with and ends with .
There are possibilities for the first digit in total, and only that works, so the probability the number begins with is
There are possibilities for the last digit, and only that works , so the probability the number ends with is
Since these are independent events, the probability both happens is
23.If each student has classes, and there are students, then they have a total of classes among them.
Each class has students, so there must be classes. Each class has teachers, so the teachers have a total of classes among them.
Each teacher teaches classes, so if there are teachers, they have classes among them. This was found to be , so
This is answer choice
24.Let the number in the top circle be and then , , , , and , going in clockwise order. Then, we have
Adding these equations together, we get
where the last step comes from the fact that since , , , , , and are the numbers in some order, their sum is
The left hand side is divisible by and is divisible by , so must be divisible by . The largest possible value of is then , and the corresponding value of is , which is choice .
It turns out this sum is attainable if you let
25.
Logically, Jane's statement is equivalent to its contrapositive,
If an even number is not on one side of any card, then a vowel is not on the other side.
For Mary to show Jane wrong, she must find a card with an odd number on one side, and a vowel on the other side. The only card that could possibly have this property is the card with , which is answer choice
Using the answer choices, we see that P and Q are logically equivalent (both are non vowel letters) and so are and (both are even numbers). Thus, if one of these are correct, then the other option in its pair must also be correct. The only remaining answer choice is
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