♣x = x² – 1
| (♣3)(♣4) | ♣11 |
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Solution
The symbol indicates the operation that must be carried out on the numbers. So for any number after this symbol we take its square and subtract one.
So for the left side we have (3² -1)(4² - 1) = 8 x 15 = 120
On the right side we have (11² -1) = 121 -1 = 120
Figure not to scale.
Angles QPS and QRS are right angles
QR = 3 , RS = 4, and PS = 2
| Side PQ | 5 |
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Solution
A line QS (not shown) divides the quadrilateral into two right angle triangles. Taking one of these triangles, QRS, we see that it is a 3-4-5 triangle with hypotenuse 5.
Now in triangle QSP, the hypotenuse is 5, and side PQ must be less than 5. (No need to calculate the exact value.)
xy ≠ 0
| x²y² | ( – xy)² |
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Solution
Since the numbers are being squared, we can ignore the minus sign. (All squares are positive.) (xy)² is the same as x²y²
r is the radius of circle C.
s is the circumference of circle C.
| 6r | s |
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Solution
The circumference of a circle is 2πr. Since π is more than 3, this makes the circumference greater than 6r.
28 percent of the students in Class A study German, 14 percent in Class B study German
| The percentage of Class A and B combined that study German | 20 percent |
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Solution
The problem here is that we do not have the totals for each class.
You cannot assume that the sizes of the classes are equal.
If they are both equal (say 100 each), then the combined percentage is 21.
But we can make the percentage less than 20 by making class B much larger than class A.
A 10-foot plank of wood is cut to give three equal lengths with a shorter length left over.
| The length of one of the equal pieces | 3 feet |
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Solution
The length of the equal parts could be 3, in which case the left over part will be 1 foot long. But there is nothing to say that the equal parts had to be integer values. If, for example, the equal parts were 3.1. feet each, then the leftover part would be 0.7 feet.
In other words we cannot be sure which is greater, and the answer is D.
The lengths of two sides of a triangle are 4 and 6
| The length of the third side | 10 |
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Solution
There is a rule you should know that is needed here.
The length of the third side of a triangle will lie between the sum and the difference of the other two sides.
In this case 2< x < 10, and so the third side will have to be les than 10.
| The sum of all the positive integer divisors of 15 | 8 |
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Solution
The positive integer divisors of 15 are 1, 15, 3, and 5. (Their sum is 24)
Do not forget that 1 and the number itself are divisors of any number.
Note that the question did not ask for prime divisors - if it had then the sum would have been 8.
The profit from a business is to be divied in the ratio 4 to 5 between X and Y respectively.
| The money X receives when the profit is $153 | $70 |
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Solution
To divide into the ratio 4 : 5 , first add the parts of the ratio (4 + 5 =9)
Now divide the money by 9 to get one part of the ratio (153/9 = 17)
To find X's share multiply by 4 (17 x 4 = 68)
Let a¤b be defined as a² – b + a
| 2¤3 | 3 |
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Solution
The strange sign tells you what operation to carry out on the numbers. Following the instructions, we have 2² - 3 + 2 on the left, which is equal to 3.