This question asks you to compare the sizes of the numbers 12/25, 0.46, and 47% and write them in ascending order.
While you could use your calculator to make them all decimal or all percentages, one elegant way to answer without using a calculator is to convert the numbers to fractions with common denominators.
To do this, multiply 12/25 by 4/4 to obtain 48/100.
Write 0.46 as 0.46/1 and multiply it by 100/100. It becomes 46/100.
Write 47% as the fraction 47/100. Now you can clearly see the size of each number compared to other numbers. In ascending order: 0.46 = 46/100, 47% = 47/100, and 12/25 = 48/100. You would then write 0.46 < 47% < 12/25.
This question says that Mahendra and Jaya shared $7224 in the ratio 7:5. You are asked to calculate how much more money Mahendra received than Jaya.
This is a Ratio and Proportion question. Mahendra received 7 parts, and Jaya received 5 parts of this money. Therefore, there are a total of 7 + 5 = 12 parts. You will need to know how much money Mahendra and Jaya each received, and then subtract them to find out how much more money Mahendra received than Jaya.
Divide $7224 by 12 to get the amount of money in each part. This is $602.
Since there are $602 in each part and Mahendra received 7 parts, multiply $602 by 7 to get the amount of money that Mahendra received. Mahendra received $602 multiplied by 7, which is equal to $4214.
Since there are $602 in each part and Jaya received 5 parts, multiply $602 by 5 to get the amount of money that Jaya received. Jaya received $602 multiplied by 5, which is equal to $3010.
$4214 - $3010 = $1204. Therefore, Mahendra received $1204 more than Jaya.
Question 1 c.
You are to calculate the expected population of Portmouth in 2030. If 550,000 is increased by 42%, you should find 42% of 550,000, then add the result to 550,000. You could also do the multiplication 550,000 × 1.42. This is the same as 550,000 × (1 + 0.42) = 550,000*1 + 550,000*0.42.
Now, 42 percent means 42 out of 100. 42% of 550,000 is 42/100 multiplied by 550,000 = 231,000. This represents the additional number of people in Portmouth. Adding it to 550,000 will give the expected population. 550,000 + 231,000 = 781,000. This is the expected population of Portmouth in 2030.
Here, you are presented with a straight-line graph with US$ on the vertical axis and EC$ on the horizontal axis. You are asked to convert $2.00 US to EC dollars. Look at the graph, at $2.00 US on the vertical axis, and follow the straight horizontal line from there to where that line intersects the graph. Now follow the vertical line from that point to the horizontal axis and see what EC$ it intersects with. You will see that the point of intersection of this vertical line and the horizontal axis is between 5 and 6 EC$. There are five small squares between 5 and 6. That means, each small square is 0.2 units wide. There are two small squares from 5, so it is at 5 + 0.2×2 = 5 + 0.4 = $5.40 EC.
You are to convert $70 EC to US$. The graph ends at $10 EC, so you cannot do this conversion by looking at and reading the value from the graph. What you can do is to find the equation of the line and then substitute $10 EC into it to find the answer.
The general equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. The y-intercept, in this case, is zero, so the equation reduces to y = mx. The gradient can be found from the graph. You could use the rise/run calculation, or you could use two points to calculate the gradient.
Suppose you use the two points (0, 0) and (8, 3) as you can see on the graph. Use the formula (y2 – y1)/(x2 – x1), where x1 = 0, y1 = 0, x2 = 8, and y2 = 3. The gradient works out to be (3 – 0)/(8 – 0) = 3/8. The equation is y = (3/8)x. Let x = 70. y = 3/8 × 70 = $26.25.
This question states that Laura uses 60 m of mesh. The length of the rectangular seedbed is 3x.
The perimeter of the rectangular seedbed is x + x + 3x + 3x = 8x, from the measurement of the left,
right, top, and bottom of the rectangle in the diagram. The perimeter of the square seedbed is
y + y + y + y = 4y, because each side is the same length. The total length of mesh needed is 60 m,
which is 8x + 4y. Therefore, 8x + 4y = 60. Dividing by 4 gives
2x + y = 15.
The question states that the areas of the rectangle and square seedbed are equal to each other. The dimensions
of the rectangular seedbed are x and 3x, so its area is x × 3x = 3x².
The area of the square seedbed is y². Therefore, y² = 3x².
So, you have two equations to work with, they are:
2x + y = 15 (Equation 1)
y² = 3x² (Equation 2)
Transpose Equation 1 to make y the subject:
y = 15 – 2x
Substitute this for y into Equation 2:
(15 – 2x)² = 3x²
(15 – 2x)(15 – 2x) = 3x²
Multiplying the brackets:
225 – 30x – 30x + 4x² = 3x²
225 – 60x + 4x² = 3x²
Subtracting 3x² from both sides:
225 – 60x + 4x² – 3x² = 0
x² – 60x + 225 = 0
You are asked to solve x² – 60x + 225 = 0 using the quadratic formula.
The quadratic formula is in the list of formulae on the CSEC Mathematics question paper. You must be careful to
correctly identify a, b, and c. In this case,
a = 1 (not x), b = -60 (not 60),
c = 225. Substitute these into the quadratic formula. Note that in substituting for b,
the quadratic equation says “-b” so you end up with -(-60) = 60.
You should obtain 55.98 and 4.02. 55.98 is too large to be x,
so the answer is 4.02 m.
You are to calculate the total area of the seedbeds. You already have x = 4.02. You are to find
y. Using y = 15 – 2x from above, y = 15 – 2×4.02 = 6.94.
x = 4.02 and y = 6.94.
The area of the rectangular seedbed is 3x² = 3×4.02² = 48 square metres.
The area of the square seedbed is 6.94² = 48 square metres.
The total area of the two seedbeds is 48 + 48 = 96 square metres.
You are to calculate the height of the flagpole FP. FP is on the right of triangle GFP, while angle FGP is on the left.
Therefore, FP is on the opposite side to FGP.
Sine FGP = sine 30 degrees = PF/PG.
PG is the hypotenuse and is 18 m long.
Sine 38 degrees = PF / 18. Therefore PF = 18 × sine of 38 degrees = 11.08 m.
Now that you have the length of PF, you can find the length of PR. In triangle PRF, PF = 11.08 m, RF = 9.7 m.
Angle PFR is 90 degrees.
Using Pythagoras’s theorem:
PR² = RF² + PF²
PR² = 9.7² + 11.08² = 216.8564
PR = √216.8564 = 14.73 m
In this question, you are presented with a diagram and are asked to find the value of x.
Angle SLQ + angle x = 180 degrees. Since PLQ is a straight line, STR = SLQ.
If you find STR, then you know SLQ, then you subtract SLQ from 180 degrees to obtain angle x.
The question states that LTR is an isosceles triangle. Therefore, two of the angles in this triangle are equal to each other.
If there are two angles equal to 28 degrees, then 28 + 28 = 56 degrees, and the third angle would be 180 – 56 = 124 degrees.
This does not seem to be the case in this diagram, so LTR is the only angle that is 28 degrees.
The other two angles are RLT and LTR.
Let y = RLT and LTR.
Angles y + y + 28 = 180 degrees because all three angles add up to 180 degrees.
2y = 180 - 28 = 152 degreesy = 152 / 2 = 76 degreesx = 180 – 76 = 104 degrees.
This can be seen to be a reflection in the line y = x.
The line y = x is a straight line that has points (0, 0), (1, 1), (2, 2), and so on.
If you plot this straight line, you will see that it is the mirror line of this reflection.
The shape, T, undergoes a translation of (-1, 6).
This means that each point on T is shifted one place in the x-direction, that is, to the left, and six places in the y-direction, that is, upwards.
The vertex (2, 1) will be shifted to (2 – 1, 1 + 6) = (1, 7).
The vertex (5, 1) will move to (5 – 1, 1 + 6) = (4, 7).
The vertex (4, 3) will move to (4 – 1, 3 + 6) = (3, 9).
The vertex (5, 3) will move to (5 – 1, 3 + 6) = (4, 9).
(1, 7), (4, 7), (3, 9), and (4, 9).
You are to calculate the length of the line PR. Sketch a graph of PQRS. You are given only the coordinates of the end points of the diagonal.
Form a right-angled triangle. You will see that the horizontal side is (4 - -3) = 7 units long and the vertical side is (10 - -4) = 14 units long.
The diagonal, PR, is the hypotenuse of the right-angled triangle.
PR² = 7² + 14² = 49 + 196
PR = √(49 + 196) = 15.65 square units.
You are to determine the equation of the line PR. The general equation of a straight line is y = mx + c, where m is the gradient of the line and c is the y-intercept.
You are given the coordinates of two points on the line, so you can determine the gradient by using the equation:
Gradient = (y₂ – y₁)/(x₂ – x₁).
If (-3, 10) is point 1, then x₁ = -3 and y₁ = 10. If (4, -4) is point 2, then x₂ = 4 and y₂ = -4.
The gradient is therefore:
(-4 – 10)/(4 - -3) = -14/7 = -2.
The equation of the line PR is of the form y = -2x + c.
You now need to find the y-intercept, c. Any point on the line can be used to find c.
If you use the point (-3, 10), then you should substitute x = -3 and y = 10 to get:
10 = -2 × -3 + c
c = 10 - 6 = 4.
The equation of the line is y = -2x + 4.
This question is under the topic Functions and Relations. You are asked to find f(x - 2).
You are told that f(x) = 3x + 1. You need to change (x) to (x - 2).
f(x - 2) = 3(x - 2) + 1 = 3x - 6 + 1 = 3x - 5.
You are required to find g(3x + 2) + 10. You need to change x to (3x + 2) in g(x) = x², and add 10.
g(3x + 2) + 10 = (3x + 2)² + 10
= (3x + 2)(3x + 2) + 10
= 9x² + 12x + 4 + 10 = 9x² + 12x + 14.
The explanation is also in the video linked above. The mode is the value that has the highest frequency. In this case, the mode is 5. Note that it is not 8 because 8 is the frequency. The mode should be the value that occurs 8 times.
The median is the value that appears at the center of the distribution after you place them in ascending or descending order. If there is an even number of values, then the median is the mean of the two values at the center of the distribution. The values are already in ascending order, from 5 to 10.
The total number of words is 8 + 4 + 2 + 2 + 3 + 4 = 23. 23/2 + 0.5 = 11.5 + 0.5 = 12. The value at the 12th position is the median. That happens to be 6. The median value is therefore 6 words.
The mean number of words is the total number of words divided by the frequency.
Note that where the number of words is 5 and the frequency is 8, the total number of words is 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40. 5*8 = 40. You must multiply the number of words by their frequency and add them up. You should then divide the result by the total frequency of 23.
The mean number of words is (5*8 + 6*4 + 7*2 + 8*2 + 9*3 + 10*4)/23 = 161/23 = 7. The mean number of words is therefore 7.
You need to calculate the number of students who took at most, 32 minutes to travel to school. "At most 32 minutes" means 32 minutes is the most. In other words, the time must not pass 32 minutes. You need to look at the horizontal axis and identify the point at which the time is 32 minutes. Note that there are five small squares between 0 and 10 on this axis. This means that each small square represents 2 units, that 10/5 = 2. Look at the 30 minute mark, then go one small square to the right to reach 32 minutes. From this point, project vertically upwards to the cumulative frequency curve, then horizontally to the left.
The vertical axis has a scale of 1 cm = 20 units. There are 5 small squares between each cm. Each small square is therefore 20/5 = 4 units. The horizontal line from the curve is between 40 and 60. From looking, this intersection seems to occur between the third and fourth lines, that is, between 52 and 56. Estimate that it is 54. Therefore, 54 students took, at most, 32 minutes to travel to school.
You are asked to determine the interquartile range. The cumulative frequency is divided into quartile, that is, four parts. 0% to 25% is the first quartile, 25% to 50% is the second quartile, 50% to 75% is the third quartile, and 75% to 100% is the fourth quartile. The interquartile range is between 25% and 75%.
The cumulative frequency has a maximum value of 220, which is 100% of its value. 25% of 220 is 25/100 × 220 = 55. 75% of 220 is 75/100 × 220 = 165. The interquartile range is 165 - 55 = 110.
There is a total of 10 letters in the given word. There are 3 "I." Therefore, the probability of picking an "I" is 3/10. "With replacement" means that Dacia put back the "I" that she picked out of the letters. The probability of getting a "V" on the second pick is 1/10 because there is only 1 "V" and 10 letters are available.
This is a situation where Dacia picks an "I" and a "V." Therefore, you must multiply the two probabilities. 3/10 * 1/10 = 3/100.
The video that is linked above explains the solution to this question. However, the explanation is also presented here in writing, for your benefit.
You are asked to calculate the shaded area of a trapezoidal prism. Its volume is 2866 cubic cm.
The volume of a uniform solid, such as this one, can be calculated by multiplying the uniform area by the length. The length of the prism is 31.2 cm.
Volume = area × length
2886 = area × 31.2.
Area = 2886 / 31.2 = 92.5 square cm.
You are presented with a cuboid-shaped bar of width 30.6 cm, length 8.2 cm, and unspecified height to be calculated. It has the same volume of 2886 cubic cm, as the trapezoidal prism in part (i).
30.6 × 8.2 × h = 2886
h = 2886 / (30.6 × 8.2) = 2886 / 250.92 = 11.502 cm.
The height is 11.5 cm.
The bar is melted down and is used to make six identical spheres. You are asked to calculate the radius of each sphere.
The volume of each sphere is 2886 / 6 = 481 cm³.
Transposing the given formula to make r the subject, and substituting the values:
r = cube root of (481 × 3 / 4 × pi) = 4.86 cm.
The surface area of each sphere is given by:
Surface area = 4 × pi × 4.86² = 296.88 cm².
You are to calculate the mass of each golden sphere, given that the density of gold is 19.3 g/cm³.
Using the formula Mass = density × volume:
Mass = 19.3 × 481 = 9283.3 grams.
Please see my explanation of the solutions to this question in the video linked above. I think the written explanation below, in combination with the video, may benefit some persons.
In this question, you are presented with four diagrams, each showing a sequence of regular hexagons. You are asked to complete the fifth diagram in the sequence.
You will notice that:
Now, you need to create Diagram 5. They presented you with an incomplete Diagram 4. To complete it:
This will make a total of 11 hexagons.
Note that in the table that you are supposed to fill out, the column that displays the number of hexagons shows 3, 5, 7, 9. You have to add two to each number to get the new number. The next number is therefore 9 + 2 = 11.
Note that in the diagrams:
Diagram n will have n rows of 2 hexagons stacked on top of the other, plus 1 on the right: 2n+1
You should write 2n + 1 as the answer in the last column below the one labeled "Number of Hexagons."
Since there are 47 in the seventh row below the heading labeled "Number of hexagons," 2n + 1 = 47.
n = (47 - 1)/2 = 46/2 = 23.
You should write the answer as 23 in the seventh column below the one labeled "Diagram Number (D)."
You now need to calculate the number of sticks.
Diagram 1: How did they get 15 sticks? There are 3 hexagons. Each hexagon had 6 sticks before you joined them. There were therefore 3 * 6 = 18 sticks. After you join the hexagons, you cover up 3 of the sticks in the joint, so you can remove those sticks. You therefore end up with 3 * 6 - 3 = 15 sticks.
Diagram 2: How did they get 23 sticks? There are 5 hexagons. Each hexagon had 6 sticks before you joined them. There were therefore 5 * 6 = 30 sticks. After you join the hexagons, you cover up 7 of the sticks in the joint, so you can remove those sticks. You therefore end up with 5 * 6 - 7 = 23 sticks.
Diagram 3: How did they get 31 sticks? There are 7 hexagons. Each hexagon had 6 sticks before you joined them. There were therefore 7 * 6 = 42 sticks. After you join the hexagons, you cover up 11 of the sticks in the joint, so you can remove those sticks. You therefore end up with 7 * 6 - 11 = 31 sticks.
Diagram 4: How did they get 39 sticks? There are 9 hexagons. Each hexagon had 6 sticks before you joined them. There were therefore 9 * 6 = 54 sticks. After you join the hexagons, you cover up 15 of the sticks in the joint, so you can remove those sticks. You therefore end up with 9 * 6 - 15 = 39 sticks.
Diagram 5: You should calculate the number of sticks. There are 11 hexagons. Each hexagon had 6 sticks before you joined them. There were therefore 11 * 6 = 66 sticks. After you join the hexagons, you cover up 19 of the sticks in the joint, so you can remove those sticks. You therefore end up with 11 * 6 - 19 = 47 sticks.
The row marked (ii) requires that you find the diagram number for 47 hexagons. Look carefully at the column with the number of hexagons, you will see that a pattern formed down the column requires that you multiply the diagram number by 3, then subtract one less than the column number to obtain the number of hexagons.
n * 3 - (n-1) = 47.
Expand the brackets:
3n - n + 1 = 47
Subtract n from 3n and subtract 1 from both sides of the equation:
2n = 46
Divide both sides of the equation by 2:
n = 23
The diagram number in the row marked "(ii)" is 23.
At the intersecting cell of the perimeter column and the row labeled "ii," you are to write the perimeter when there are 47 hexagons. The perimeter for a single hexagon is 6 because it has 6 sides. When you join two hexagons at one side, one of the sticks can be removed if you are counting the number of sticks. However, if you are considering the perimeter, then two sticks, one on each hexagon, are not considered. Considering this fact, the pattern of the numbers under perimeter, along with the formula for the number of hexagons and the number of sticks, you can say that the perimeter of each figure is (the number of hexagons multiplied by 6) minus (the figure number multiplied by 3 times 2).
In the row labeled "(iii)," you are to write down each formula in terms of n. Considering the instructions in the linked video above and the arguments about how you obtain the figures in each column, the formulas simplify to:
This question requires that you show why Skyla is incorrect in saying that she can make one of these figures with a perimeter exactly 1005. To do so, you must equate the formula for the perimeter to 1005, then solve for n.
The formula for the perimeter of the figure is:
Perimeter = 4n + 8
Now, equate the formula to 1005:
4n + 8 = 1005
Solving for n:
Since n must be a positive integer for Skyla to be correct, and 249.25 is not a positive integer, we conclude that Skyla is incorrect.