UPCAT Simulated Exam Math Quiz!
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Questions and Answers
- 1.
A circular table is pushed into a corner of a rectangular room so that it touches both walls. A point on the edge on the edge of the table between the two points of contact is 2 inches from one wall and 9 inches from the other wall. What is the radius of the table?
- A.
20 inches
- B.
12 inches
- C.
Approximately 15 inches
- D.
Approximately 9.22 inches
Correct Answer
D. Approximately 9.22 inchesExplanation
We have a right triangle with sides of 2 inches and 9 inches, and we’re solving for the hypotenuse, which is the radius of the table.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
a2+b2=c2
In this case, a is 2 inches, b is 9 inches, and c is the radius of the table we’re trying to find.
So, we can set up the equation as follows:
22+92=r2
4+81=r2
85=r2
Taking the square root of both sides to solve for r gives us:
√r2=√85​
To get the decimal approximation of the square root of 85, we can use a calculator. The square root of 85 is approximately 9.219544457292887.
So, when rounded to two decimal places, the radius of the table is approximately 9.22 inches.Rate this question:
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- 2.
Twelve consecutive integers are added together. What is the remainder when the sum is divided by 4?
- A.
1
- B.
2
- C.
3
- D.
No remainder
Correct Answer
B. 2Explanation
To find the remainder when the sum of twelve consecutive integers is divided by 4, we can start by denoting the first integer as n. The twelve consecutive integers would then be:
n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8, n+9, n+10, n+11
Next, we can calculate the sum of these integers:
Sum = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) + (n+6) + (n+7) + (n+8) + (n+9) + (n+10) + (n+11)
This simplifies to:
Sum = 12n + (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)
The sum of the integers from 0 to 11 is:
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66
So, we can rewrite the total sum:
Sum = 12n + 66
Next, we need to find the remainder of this sum when divided by 4. To do that, we will evaluate each part of the expression modulo 4.
12n mod 4: Since 12 is a multiple of 4, 12n mod 4 = 0 for any integer n.
66 mod 4: Now we calculate the remainder of 66 when divided by 4:
66 divided by 4 equals 16.5 (the integer part is 16)
16 times 4 equals 64
66 minus 64 equals 2
Thus, 66 mod 4 = 2.
Now, combining the two parts:
(12n + 66) mod 4 = (0 + 2) mod 4 = 2
Therefore, the remainder when the sum of twelve consecutive integers is divided by 4 is:
2Rate this question:
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- 3.
Find the total surface area in square meters of a rectangular solid whose length is 7 meters, width is 6 meters, and depth is 3 meters.
- A.
32
- B.
162
- C.
81
- D.
126
Correct Answer
B. 162Explanation
The total surface area of a rectangular solid can be found by adding up the areas of all six faces. In this case, the length is 7 meters, the width is 6 meters, and the depth is 3 meters. The formula for the surface area of a rectangular solid is 2lw + 2lh + 2wh. Plugging in the given values, we get 2(7)(6) + 2(7)(3) + 2(6)(3) = 84 + 42 + 36 = 162 square meters.Rate this question:
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- 4.
A cube has a volume of 125cm3. What is the area of one face of a cube?
- A.
20 sqcm
- B.
25 sqcm
- C.
41.33 sqcm
- D.
5 sqcm
Correct Answer
B. 25 sqcmExplanation
The volume of a cube is calculated by multiplying the length, width, and height of the cube. In this case, the volume is given as 125cm3. Since all sides of a cube are equal, we can find the length of one side by taking the cube root of the volume. The cube root of 125 is 5. Therefore, the length of one side of the cube is 5cm. The area of one face of the cube is found by squaring the length of one side. The square of 5 is 25. Therefore, the area of one face of the cube is 25 sqcm.Rate this question:
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- 5.
The diagonals of a rhombus measure 24 inches and 32 inches. The perimeter of a rhombus.
- A.
80 inches
- B.
160 inches
- C.
100 inches
- D.
180 inches
- E.
Cannot be determined uniquely from the given information
Correct Answer
A. 80 inches -
- 6.
A car traveled 3 miles south, 9 miles east, and then another 9 miles south. What is the straight-line distance (in miles) between the point where the car started and where it stopped?
- A.
21
- B.
18
- C.
3
- D.
15
Correct Answer
D. 15Explanation
The car first traveled 3 miles south, then 9 miles east, and finally another 9 miles south. If we draw a straight line from the starting point to the stopping point, it would form a right-angled triangle. The distance traveled south can be considered as the height of the triangle, which is 3 + 9 = 12 miles. The distance traveled east can be considered as the base of the triangle, which is 9 miles. Using the Pythagorean theorem, we can calculate the hypotenuse (the straight-line distance) as the square root of (12^2 + 9^2) = 15 miles. Therefore, the correct answer is 15.Rate this question:
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- 7.
The diagonals of a rhombus measure 24 inches and 32 inches. The perimeter of a rhombus
- A.
Cannot be determined uniquely from the given information
- B.
113 inches
- C.
80 inches
- D.
60 inches
Correct Answer
C. 80 inchesExplanation
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- 8.
If f(x) = 3x^2 - 4x + 5, what is the value of f'(2)?
- A.
8
- B.
12
- C.
10
- D.
2
Correct Answer
A. 8 -
- 9.
The sum of the first five terms of a geometric sequence is 93. If its common ratio is 2, what is the first term?
- A.
2
- B.
4
- C.
3
- D.
5
Correct Answer
C. 3Explanation
The sum of the first five terms of a geometric sequence can be found using the formula S = a(r^n - 1) / (r - 1), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, we are given that S = 93, r = 2, and n = 5. Plugging these values into the formula, we can solve for a.Rate this question:
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- 10.
A certain number was to be added to 1/2 but by mistake 1/2 was divided by the number. However, the correct result was obtained. Find the number.
- A.
2 or -1
- B.
-2 or 1
- C.
1/2 or -1
- D.
1/4 or 1
Correct Answer
C. 1/2 or -1 -
- 11.
The sides of a right triangle are denoted as x, (x + 1) and (x + 2). What is the largest side of the triangle?
- A.
5
- B.
6
- C.
3
- D.
4
Correct Answer
A. 5 -
- 12.
A letter is chosen at random from the word “superstar”. What is the probability that a letter is a vowel?
- A.
2/3
- B.
1/3
- C.
1/9
- D.
1/6
Correct Answer
B. 1/3Explanation
The word "superstar" has 9 letters. Out of these 9 letters, there are 3 vowels (u, e, a). Therefore, the probability of randomly choosing a vowel from the word "superstar" is 3/9, which simplifies to 1/3.Rate this question:
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- 13.
The measures of the sides of a triangle are 10, 13, and 13 cm, how far is the shortest side to the opposite vertex?
- A.
11 cm
- B.
12 cm
- C.
12.5 cm
- D.
14
Correct Answer
B. 12 cm -
- 14.
What is 2563/4?
- A.
192
- B.
64
- C.
128
- D.
160
Correct Answer
B. 64 -
- 15.
A store sells calamansi at 8 for P2.00. If it changes the price to 10 calamansi for P3.00, what is the percent increase in price per calamansi?
- A.
20%
- B.
25%
- C.
33 1/3%
- D.
40%
Correct Answer
A. 20%Explanation
The original price of calamansi was 8 for P2.00, which means each calamansi cost P0.25. After the price change, the new price is 10 calamansi for P3.00, which means each calamansi now costs P0.30. To find the percent increase, we calculate the difference between the new and old prices, which is P0.30 - P0.25 = P0.05. Then we divide this difference by the old price, P0.05 / P0.25 = 0.2. Finally, we multiply this result by 100 to find the percentage increase, 0.2 * 100 = 20%.Rate this question:
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- 16.
The minute hand of a clock has made sixteen complete revolutions since noon, what angle (in degrees) does the minute hand make with the hour hand?
- A.
60
- B.
150
- C.
90
- D.
120
Correct Answer
D. 120Explanation
The minute hand of a clock makes one complete revolution every 60 minutes. Since it has made sixteen complete revolutions since noon, it means that 16 hours have passed. The hour hand moves 30 degrees per hour, so in 16 hours, it has moved 16 * 30 = 480 degrees. The minute hand moves 360 degrees in one complete revolution. Since the hour hand is at 480 degrees and the minute hand has made 16 complete revolutions, it means that the minute hand is at 480 + (16 * 360) = 6120 degrees. The angle between the minute hand and the hour hand is the difference between their positions, so it is 6120 - 480 = 5640 degrees. However, we only need to consider the angle within one complete revolution, so we subtract 360 degrees from 5640, resulting in 5280 degrees. Finally, we subtract 5280 degrees from 6120 degrees to find the angle between the minute hand and the hour hand, which is 6120 - 5280 = 840 degrees. However, we only need to consider the smaller angle within one complete revolution, so we subtract 360 degrees from 840, resulting in 480 degrees. Therefore, the angle the minute hand makes with the hour hand is 480 degrees.Rate this question:
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- 17.
Solve for x: 4x = 8
- A.
1/2
- B.
3
- C.
-2
- D.
3/2
Correct Answer
D. 3/2Explanation
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- 18.
What are the roots of the equation x2 + 3x - 10
- A.
-5, 2
- B.
3, 1
- C.
-2, 5
- D.
5, 2
Correct Answer
A. -5, 2 -
- 19.
What is (1 + 2)2 + (3 - 1) * 33 + 1?
- A.
64
- B.
3
- C.
6
- D.
-3
Correct Answer
A. 64Explanation
The expression is (1 + 2)² + (3 - 1) * 3³ + 1.Evaluate inside the parentheses first:
(1 + 2)² = (3)² = 9
(3 - 1) = 2Evaluate the exponent:
(3)² = 9Evaluate the exponentiation:
(3³) = 27Evaluate the multiplication:
(2) * (27) = 54Add the results together:
9 + 54 + 1 = 64Rate this question:
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- 20.
Which of the following statement is not true?
- A.
The slope of a vertical line is undefined
- B.
The slope of a line from left to right, going upwards is positive
- C.
The slope of a line from left to right, going downwards is negative
- D.
None of the above
Correct Answer
D. None of the aboveExplanation
The statement "none of the above" is not true because all of the statements given are true. The slope of a vertical line is undefined because it does not have a defined rise over run. The slope of a line from left to right, going upwards is positive because it has a positive rise over run. The slope of a line from left to right, going downwards is negative because it has a negative rise over run. Therefore, all of the statements given are true.Rate this question:
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- 21.
For every negative number z, if x > y, then
- A.
X>z
- B.
Xz
- C.
X+z
- D.
None of the above
Correct Answer
D. None of the aboveExplanation
The given statement does not provide enough information to determine the relationship between x and z. Therefore, none of the options can be definitively concluded based on the given information.Rate this question:
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- 22.
The second and fifth term of a geometric progression is 6 and 162 respectively. What is the common ratio?
- A.
1/2
- B.
1/3
- C.
3
- D.
2
Correct Answer
C. 3Explanation
In a geometric progression, each term is obtained by multiplying the previous term by a constant called the common ratio. In this case, we can find the common ratio by dividing the fifth term (162) by the second term (6). 162 divided by 6 equals 27. Therefore, the common ratio is 27.Rate this question:
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- 23.
If m, m+3, 2m-5 is an arithmetic progression, what is the value of m?
- A.
3
- B.
8
- C.
14
- D.
11
Correct Answer
D. 11 -
- 24.
What is the sum of the first 40 multiples of 4?
- A.
3200
- B.
3280
- C.
6560
- D.
6400
Correct Answer
B. 3280Explanation
The sum of the first 40 multiples of 4 can be found by using the formula for the sum of an arithmetic series. The formula is given by S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term. In this case, the first term is 4 and the last term is 160. Plugging these values into the formula, we get S = (40/2)(4 + 160) = 20(164) = 3280. Therefore, the correct answer is 3280.Rate this question:
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- 25.
If 1 < a < 5 and 2 < b < 7, what is the greatest value of a2 - b2?
- A.
1
- B.
24
- C.
48
- D.
21
Correct Answer
D. 21 -
- 26.
If the sum of the squares of two consecutive odd integers is 290, what are the integers?
- A.
11 and 13
- B.
13 and 15
- C.
15 and 17
- D.
17 and 19
Correct Answer
A. 11 and 13 -
- 27.
What is the x-intercept of the line 2x + 7y +15 = 0?
- A.
15/7
- B.
-15/2
- C.
-15
- D.
2/15
Correct Answer
B. -15/2Explanation
The x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept, we set y=0 in the equation of the line and solve for x. In this case, when we substitute y=0 into the equation 2x + 7y + 15 = 0, we get 2x + 15 = 0. Solving for x, we find x = -15/2. Therefore, the x-intercept of the line is -15/2.Rate this question:
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- 28.
In how many ways can four students sit in a row of four chairs?
- A.
24
- B.
16
- C.
4
- D.
1
Correct Answer
A. 24Explanation
There are four students and four chairs, so the first student has four choices for a seat. After the first student is seated, there are three remaining seats for the second student to choose from. Once the second student is seated, there are two remaining seats for the third student, and finally, the last student has only one seat left to choose from. Therefore, the total number of ways the four students can sit in a row of four chairs is 4 x 3 x 2 x 1 = 24.Rate this question:
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- 29.
How many 3-digit numbers can you get from the digits 1, 2, 3, 4, 5 without repeating the digits?
- A.
15
- B.
60
- C.
120
- D.
240
Correct Answer
B. 60Explanation
To find the number of 3-digit numbers that can be formed without repeating the digits, we need to calculate the number of permutations. We have 5 digits to choose from for the hundreds place, 4 digits for the tens place (as one digit has already been used), and 3 digits for the units place. Therefore, the total number of permutations is 5 x 4 x 3 = 60. Hence, the correct answer is 60.Rate this question:
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- 30.
There are 12 teams in the Asian Cup of the football game. If each team must play once with all the other teams, how many games should be played?
- A.
24
- B.
66
- C.
132
- D.
144
Correct Answer
B. 66 -
- 31.
What is the probability of getting a sum of 10 from rolling a pair of 6 sided dice?
- A.
1/2
- B.
1/12
- C.
1/13
- D.
1/36
Correct Answer
B. 1/12Explanation
When rolling a pair of 6-sided dice, there are a total of 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die). To get a sum of 10, there are only 3 possible outcomes: (4,6), (5,5), and (6,4). Therefore, the probability of getting a sum of 10 is 3/36, which simplifies to 1/12.Rate this question:
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- 32.
In a regular deck of 52 cards, a face card is a jack, a queen, or a king. What is the probability of picking a face card from a regular deck?
- A.
1/2
- B.
1/4
- C.
1/13
- D.
3/13
Correct Answer
D. 3/13Explanation
In a regular deck of 52 cards, there are 12 face cards (4 jacks, 4 queens, and 4 kings). The probability of picking a face card from a regular deck can be calculated by dividing the number of face cards by the total number of cards in the deck. Therefore, the probability is 12/52, which simplifies to 3/13.Rate this question:
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- 33.
There are 100 ping pong balls numbered 1 to 100 in a dropbox. If a ball is randomly picked from a dropbox, what is the probability of getting a ball divisible by 6?
- A.
3/50
- B.
1/16
- C.
4/25
- D.
1/100
Correct Answer
C. 4/25Explanation
The probability of getting a ball divisible by 6 can be calculated by finding the number of balls divisible by 6 and dividing it by the total number of balls. In this case, there are 16 balls (6, 12, 18, ..., 96, 100) that are divisible by 6, out of a total of 100 balls. Therefore, the probability is 16/100, which simplifies to 4/25.Rate this question:
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- 34.
From a set of 10 different entries in a painting contest, 3 are to be chosen to receive the first, second, and third prize. How many ways can these prizes be awarded?
- A.
5040
- B.
720
- C.
60
- D.
120
Correct Answer
B. 720Explanation
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- 35.
A 5 centavo coin and a 10 centavo coin are each tossed once and their top faces noted. What is the probability of tails on both coins?
- A.
1/2
- B.
2/3
- C.
1/4
- D.
3/4
Correct Answer
C. 1/4Explanation
When two coins are tossed, each coin has two possible outcomes: heads or tails. Since we are interested in the probability of getting tails on both coins, we need to consider the probability of getting tails on the first coin (1/2) multiplied by the probability of getting tails on the second coin (1/2). Multiplying these probabilities gives us 1/4, which is the probability of getting tails on both coins.Rate this question:
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- 36.
In how many ways can 8 different beads be put in a string to make a ring?
- A.
40320
- B.
2520
- C.
5040
- D.
8
Correct Answer
B. 2520Explanation
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- 37.
If a basketball court in the MBA is 100ft long and half of the court is an exact square, what is the area of one half-court?
- A.
1000 sqft
- B.
2500 sqft
- C.
6250 sqft
- D.
3000 sqft
Correct Answer
B. 2500 sqftExplanation
Since half of the basketball court is an exact square, we can determine the length of one side of the square by dividing the length of the court by 2. Therefore, the length of one side of the square is 100ft / 2 = 50ft. The area of a square is calculated by multiplying the length of one side by itself, so the area of the square is 50ft * 50ft = 2500 sqft. Therefore, the area of one half-court is 2500 sqft.Rate this question:
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- 38.
The width of a rectangle is 4/5 of its length. If its perimeter is 72, what is its area?
- A.
160
- B.
250
- C.
280
- D.
320
Correct Answer
D. 320Explanation
Let's assume the length of the rectangle is L. According to the given information, the width of the rectangle is 4/5 of its length, which means the width is (4/5)L.
The perimeter of a rectangle is given by the formula P = 2(L + W), where P is the perimeter, L is the length, and W is the width.
Substituting the given values, we have 72 = 2(L + (4/5)L).
Simplifying the equation, we get 72 = 2(9/5)L.
Dividing both sides by 2, we get 36 = (9/5)L.
Multiplying both sides by 5/9, we get L = 20.
Therefore, the length of the rectangle is 20 units and the width is (4/5)(20) = 16 units.
The area of a rectangle is given by the formula A = L * W.
Substituting the values, we get A = 20 * 16 = 320.
Hence, the area of the rectangle is 320.Rate this question:
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- 39.
If the height of a triangle is halved and its base doubled, what is the ratio of the area of the new triangle to the area of the old one?
- A.
2 : 1
- B.
1 : 1
- C.
1 : 2
- D.
2 : 3
Correct Answer
B. 1 : 1Explanation
When the height of a triangle is halved and its base is doubled, the area of the new triangle remains the same as the area of the old one. This is because the formula for the area of a triangle is 1/2 * base * height. When the height is halved, the factor of 1/2 is compensated by doubling the base, resulting in the same area. Therefore, the ratio of the area of the new triangle to the area of the old one is 1:1.Rate this question:
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- 40.
What is the maximum rectangular area that can be enclosed in a wire of length of 8m?
- A.
1 sqm
- B.
4 sqm
- C.
8 sqm
- D.
16 sqm
Correct Answer
B. 4 sqmExplanation
The maximum rectangular area that can be enclosed in a wire of length 8m is 4 sqm. This can be achieved by creating a square with sides measuring 2m each. The perimeter of this square would be 8m, and the area would be 4 sqm.Rate this question:
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- 41.
If the radius of a circle is increased by 30%, by how much will the area increase?
- A.
60%
- B.
120%
- C.
169%
- D.
69%
Correct Answer
D. 69%Explanation
When the radius of a circle is increased by 30%, the new radius becomes 1.3 times the original radius. The area of a circle is calculated using the formula A = πr^2, where r is the radius. If we substitute the new radius into the formula, we get A' = π(1.3r)^2 = π(1.69r^2). Comparing this to the original area, the area increase is 69% (1.69 - 1 = 0.69 or 69%).Rate this question:
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- 42.
ABCDE is a regular pentagon, what is the measure of an interior angle?
- A.
72 degrees
- B.
108 degrees
- C.
144 degrees
- D.
200 degrees
Correct Answer
B. 108 degreesExplanation
A regular pentagon has five equal sides and five equal interior angles. To find the measure of an interior angle, we can use the formula (n-2) * 180 / n, where n is the number of sides of the polygon. Plugging in the values, we get (5-2) * 180 / 5 = 3 * 180 / 5 = 540 / 5 = 108 degrees. Therefore, the measure of an interior angle of a regular pentagon is 108 degrees.Rate this question:
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- 43.
When a can of sardines sells at P16.80 each, the supermarket makes a 40% profit. What should the sale price of a can be if the store is willing to make only a 10% profit?
- A.
P 11.75
- B.
P 13.20
- C.
P 12.00
- D.
P 14.60
Correct Answer
B. P 13.20Explanation
To find the sale price of a can with a 10% profit, we need to calculate the cost price of the can. Since the supermarket makes a 40% profit when selling the can at P16.80, the cost price can be calculated as follows:
Cost Price = Selling Price / (1 + Profit %)
Cost Price = P16.80 / (1 + 40/100)
Cost Price = P16.80 / 1.40
Cost Price = P12
Now, to make a 10% profit, we can calculate the sale price as follows:
Sale Price = Cost Price + (Cost Price * Profit %)
Sale Price = P12 + (P12 * 10/100)
Sale Price = P12 + P1.20
Sale Price = P13.20
Therefore, the sale price of a can should be P13.20.Rate this question:
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- 44.
The bus fare is P20.00 for the first 5 km and P3.50 for every additional kilometer. What is the bus fare for a 45 km bus trip?
- A.
P 145
- B.
P 155
- C.
P 150
- D.
P 160
Correct Answer
D. P 160Explanation
The bus fare for the first 5 km is P20.00. For every additional kilometer after the first 5 km, the fare is P3.50. Since the bus trip is 45 km, we need to calculate the fare for the additional 40 km. 40 km multiplied by P3.50 gives us P140.00. Adding this to the initial fare of P20.00, we get a total fare of P160.00.Rate this question:
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- 45.
A man wants to cover a distance by car in 20% less time than he usually takes. By what percent must he increase his driving speed?
- A.
20
- B.
30
- C.
25
- D.
40
Correct Answer
C. 25 -
- 46.
Mr. Antonio, an Engineering teacher is writing a test consisting of a total of 30 problems worth 2-points, 3-points, and 5-points each. If the number of 2-point problems is twice the number of 5-point problems, and the number of 3-point problems, is 5 less than the number of 2-point problems, what is the total value of the test?
- A.
50
- B.
90
- C.
100
- D.
36
Correct Answer
B. 90 -
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