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mathematics
cambridge checkpoint lower secondary mathematics
Cambridge Checkpoint Lower Secondary Mathematics Student's Book 7 3rd Edition Frankie Pimentel, Ric Pimentel, Terry Wall - Solutions
Work out the following percentages. a. 50% of 300b. 75% of 400c. 40% of 600d. 25% of 360e. 80% of 250f. 12.5% of 320g. 37.5% of 240h. 87.5% of 880
Three students, Alec, Jaime and Sophia, are comparing results in their tests.Alec scored 32/40 , Jaime scored 55/70 and Sophia scored 20/26.Which student got the highest score?As the tests were out of different totals, it is difficult to compare the results as they are.A way of comparing them more
Write the first quantity as a percentage of the second.a. 36 out of 72b. 18 out of 90c. 5 out of 20
15% of students in a school of 720 students are left-handed. What number of students are right-handed?
Write the first quantity as a percentage of the second.a. 6 out of 60b. 36 out of 60c. 54 out of 60
The mass of a puppy increases from 1.25 kg to 10 kg when fully grown. What is the percentage of the adult mass compared with the puppy mass?To work this out it helps to visualize the masses.The mass has gone up by a factor of × 8 and therefore the adult mass is 800% of the puppy mass. this is
At a Cambridge college, 65% of the maths students come from England. If there are 240 maths students, how many are not from England?
a. Match each of the percentages on the left to their equivalent fractions on the right. Note, that some fractions have no equivalent percentage pair.b. For each matched pair in (a) above, calculate that amount out of 600.
Write the first quantity as a percentage of the second.a. 100 compared to 80 b. 160 compared to 80 c. 400 compared to 80
A survey showed that 75% of vehicles passing a school were cars. If 4200 vehicles passed the school in one day, how many were cars?
Write the first quantity as a percentage of the second.a. 1 out of 500 b. 1/2 out of 500 c. 500 out of 500
In a village of 3500 people, 12% are left-handed, and 15% of the left handed people have blonde hair. How many people in the village are blonde and left-handed?
A school football team plays 25 games in a season. They win 17, draw 2 and lose the rest. Express the numbers won, drawn and lost as percentages of the total number of games played.
In a class of 32 students in a school, the pupils have the option of playing football (F) and/or tennis (T) for their sports lessons.The numbers are given in the Venn diagram below.a. What percentage of students play neither sport?b. What percentage of students play tennis? F 12 8 ? T 2
a. By copying the diagram below twice, use it to show that 20% of 40 is the same as 40% of 20.b. Copy the diagram below and explain how it can be used to calculate 87.5% of 56. 4 10
A 5 × 4 grid is shown below, with two of its squares shaded.a. What percentage of the grid is shaded?b. Four of the grids are arranged together as shown.(i) Choosing from the options below, what percentage of the larger grid is now shaded?2% 8% 10% 20%
An airline has a fleet of 80 planes. Of these, 10 are being serviced at any one time. What percentage of the fleet is available for flights?
A car manufacturer produces 175000 sports cars a year. They are only available in white, black and red. 52500 are white, 78 750 are black and the rest are red.What percentage of the cars are red?
A disease affecting cats is known to exist in 0.25% of the cat population. If a vet tests 2800 cats in a year, how many cats would be expected to carry the disease?
A small town has a population of 16 200 people. 3/1% of the population are over 100 years old.How many people are over 100?
Yussef takes two maths papers for an exam.For Paper 1 he scored 55/80, whilst for Paper 2 he scored 105/120.Calculate his percentage score for the whole exam.
A farmer sells honey in jars at a market. He has 60 jars to sell.He calculates that the cost of producing each jar of honey is $1.50.He manages to sell 25 jars at $3.50 each.He then decides to reduce the price of the honey to $2.00 for each jar and as a result manages to sell the rest.a. Calculate
A school has 230 students, all of whom study maths. It is estimated that on average, each student uses two exercise books per year.The school can only buy maths exercise books in packs of 25. Assuming the school buys the minimum number of exercise books necessary for the whole year, what percentage
A woman wants to tile her bathroom. The wall to be tiled is a rectangle with dimensions 2.5m × 1.2m.The tiles she wants measure 10cm × 10cm and are only sold in boxes of 90 tiles.What percentage extra will she have to buy in order to tile her bathroom?
A student runs a 400m race in t seconds. His friend manages to run the race four seconds faster. Write an expression, in terms of t, for the time taken by the friend to run the race.
a. Derive a formula for converting the number of hours, h, into minutes, m.b. Use your formula to convert 4 1/2 hours to minutes.
In one year, one dollar ($1) could be exchanged for one euro and twenty cents (€1.20). a. Derive a formula to exchange dollars into euros.Let d be the number of dollars.Let e be the number of euros.From the information in the question, e=1.2 when d=1. So the formula is:e = 1.2db. Use
A second-hand computer game costs $x to buy. A new copy of the game costs six dollars more than double the cost of the second-hand one.a. Write an expression, in terms of x, of the cost of buying the new copy.b. If the second-hand computer game costs $5.50, calculate the cost of the new copy.
a. Derive a formula for converting the number of days, d, into hours, h.b. Use your formula to convert 14 days to hours.
An electrician charges a fee of $25 per hour.a. Derive a formula to show his charges, where C is the total charge and n is the number of hours.C = 25nb. What would the electrician charge for working 6 hours? C = 25 × 6C = 150The total charge is $150.
Isabel achieves m marks in a science test. Another student in the class, Luis, gets 15 fewer than triple the amount of marks achieved by Isabel.a. Write an expression, in terms of m, for the amount of marks achieved by Luis.b. If Isabel got 22 marks, calculate how many Luis got.
The distance between two cities is 128km. The time taken to travel between the two cities is the distance divided by the average speed (skm/h).a. Write a formula for the time (T) taken to travel between the two cities.b. Calculate the time taken to travel between the two cities if the average speed
a. Derive a formula for converting the number of hours, h, into seconds, s.b. Use your formula to convert 2 1/2 hours to seconds.c. Rearrange your formula to convert seconds into hours and use it to work out how many hours are in 27 000 seconds.
Kimie is saving some money in her bank account for a holiday. At the start of the year she has ¥x in her bank account. By the end of the year she has ¥200 less than four times the amount she had at the start of the year.a. Write an expression, in terms of x, for the amount of money Kimie has in
a. Derive a formula for converting the number of metres, m, into centimetres, c.b. Use your formula to convert 3.3 metres to centimetres.c. Rearrange your formula to convert centimetres to metres and use it to work out how many metres are in 4550cm.
Box A contains x number of sweets.Box B contains ten more than double the number of sweets as box A.Write an expression for the number of sweets in box B.2x + 10
Two rectangles A and B are shown below. a. If x = 2, which rectangle has the greatest perimeter? Justify your answer.b. If x = 5, which rectangle has the greatest perimeter? Justify your answer. X+2 3x+5 А 5x-2 B 2x
A girl is y years old. Her grandmother is four years less than treble the girl’s age.Write an expression for the age of the grandmother.3y−4
Write a mathematical problem to represent the expression 3x − 2.
Two beakers of water are shown below. If the amount of water in the first beaker is 50ml and represented by x, which of the following expressions can represent the amount of water in the second beaker? 4x + 10 3x + 605x − 40Write another expression, in terms of x, to represent the
Write a mathematical problem to represent the expression 6x + 12.
There are three jigsaw puzzle boxes. The first box has n pieces, the second has 3n pieces and the third has 2n + 40 pieces.a. Is it possible to tell which box has the least number of pieces? Justify your answer.b. Is it possible to tell which box has the most number of pieces? Justify your answer.
Four children are shown below. Their heights are given in terms of x. a. Name one child who is definitely shorter than another child. ………………. is shorter than…………………b. Which one child could be either the shortest or the tallest? Justify your answer. •F Annie: x-5
Take a drawing pin and drop it on the desk 100 times.a. Make a tally chart to record the number of drops and the number of times it lands point up.b. Use your tally chart to calculate the relative frequency and therefore estimate the experimental probability of the pin landing point up.c. Combine
a. A coin is flipped ten times and it shows heads four times. Based on this result, estimate the experimental probability of getting heads.Experimental probability is 4/10 or 0.4.b. The coin is flipped 100 times and it shows heads 42 times. Calculate the experimental probability. The experimental
a. Make a spinner out of card and divide it into four equal parts similar to that shown. Stick a pencil through its centre so that it can be spun.b. What is the theoretical probability of the spinner landing on each of the colours?c. (i) Spin your spinner 10 times and record the results in a
Flip two coins together ten times.a. Record the results in a tally chart as HH (two heads), TT (two tails) or HT (a heads and a tails in any order).b. Estimate the experimental probability of each.c. How do your results compare with theoretical probability of each outcome?d. Combine your results
An Excel spreadsheet can be made to generate random integers between certain limits. To make the spreadsheet simulate the rolling of an ordinary dice, generating random integers between 1 and 6 will be needed. a. In cell A1 type ‘=RANDBETWEEN(1,6)’ and then press return. Copy the formula
One student places a total of 50 counters of three different colours in to a bag (for example, they place 30 red, 15 blue and 5 green counters in a bag).The aim of the activity is for a second student to predict how many counters of each colour are in the bag. The second student takes a counter
Construct an equation from the information given in each question and then solve it.a. I think of a number and add 10. The answer is 24. What is the number?b. I think of a number and subtract 7. The answer is 7. What is the number?c. A number times 7 is 42. What is the number?d. I think of a number
Look at these scales.The mass of each of the small boxes is 1kg. Calculate the mass of the large box.Because the scales are balanced, the left-hand side of the scales must weigh the same as the right-hand side.In order to keep the scales balanced, whatever we do to the left-hand side of the scales
A bus has two levels. The ground level can hold a maximum of x passengers. The top floor can hold 16 passengers less than the ground floor. If the maximum number of passengers the whole bus can take is 112, calculate how many passengers each floor can have. Show your working clearly. 回
Look at these scales. The mass of each of the small boxes is 1kg. Calculate the mass of each of the large boxes.We want to find something that we can do to both sides which will leave just one large box on the left-hand side. We cannot take two large boxes from both sides as there are not two
Solve the following equations:a. 5v=20b. 6w=42c. 4x=48d. 6y=60e. 7z=49f. 14=7ag. 24=8bh. 54=9ci. 144=12d
Write down the inequalities that correspond to the following number lines: a. b. 1 2 3 4
The perimeter of the rectangle is 22 cm. Find the length of the longer side. Show your working clearly. 3x+1 4
A rectangle has dimensions as shown.Explain why if its perimeter is 24cm its area cannot be 30cm2. 2x-1 7
Write the following sentences using inequality signs:a. The maximum capacity of an athletics stadium is 20 000 people.b. Five times a number plus 3 is less than 20.c. The maximum temperature in May was 25˚C.d. A number doubled and then with 6 subtracted from it is greater than 50.
Ayse and Ahmet are playing a ‘think of a number’ game.Ayse says, ‘I think of a number, double it and subtract 5.The answer is 27.’What is the number Ayse chose?Using n to represent the number Ayse chose, we can write:2n − 5 = 27 Then we solve the equation as before: So the
Solve the following equation:3t − 4 = 8In this case, in order to get 3t on its own on the left-hand side of the equation, we need to add 4 to each side. 3t − 4 + 4 = 8 + 4 3t = 12 t =
Three containers P, Q and R each contain a different number of sweets.Container Q has five more than P. Container R has two less than double the number in P. If the total number of sweets in all three containers is 83, calculate, by forming and solving an equation, how many sweets are in each
For each of the angles shown:(i) Estimate its size (ii) Measure it and check how good your estimate was.Aim for your estimate to be within 10° of the actual angle.a.b.c.d.e.f.
Using a ruler and set squares only, construct the following shapes: a.b.c.d. 5cm 5cm
In each of the questions below calculate the size of the angles labelled with letters, giving a reason for your answer each time. Calculate the size of angle a. Justify your answer. 50° 135⁰ qº 70°
Find the size of each of the unknown angles in these diagrams: a. b. Z 35% W V y X
Find the size of the unknown angle in each of these questions: a. b. c. d. e. f. a 120⁰ 130⁰
a. Draw a similar diagram to the one shown below. Measure carefully each of the labelled angles and write them down.b. Draw a similar diagram to the one shown below. Measure carefully each of the labelled angles and write them down.c. Draw a similar diagram to the one shown below. Measure carefully
a. Draw a similar diagram to the one shown below. Measure carefully each of the labelled angles and write them down. b. Draw a similar diagram to the one shown below. Measure carefully each of the labelled angles and write them down.c. Draw a similar diagram to the one shown below. Measure
Find the size of each of the unknown angles in these diagrams: a. b. c.d. e.f. g. h. 40° b b a
Measure the angle drawn below. -Place the protractor over the angle as shown.– Align the 0˚ with one of the lines drawn– Look at the scales to see which one you will use. In this case it is the inner scale which starts at 0˚. The angle is 41˚.
Measure the angles in the diagrams below. a.b.c.d.e.f. b a
Using appropriate geometric equipment, construct the following: a. b. c. d. 6 cm 30° 6cm 6cm 6 cm
Find the size of the unknown angles in each of these questions a. b. c. d. 22° g 142°
Draw an angle of 120˚.– First draw a straight line about 6cm long.– Place the protractor on the line so that the central cross hair is on one of the end points of the line. Make sure the line lines up with the 0˚ on the protractor.– Decide which scale to use. (In this case it is the outer
Two types of set square are shown below: a. An architect says that she can draw an angle of 75° using these two set squares. Show how this can be done.b. She also says she can draw an angle of 15°. Show how this can be done.c. State two obtuse angles she could also draw using her set
In the following question:(i) Draw by estimating (without a protractor) angles of the following size(ii) Alongside your first diagram, draw the same angles with a protractor. Compare your answers. How good were your estimates?a. 20˚b. 45˚c. 90˚d. 120˚e. 157˚f. 172˚g. 14˚h. 205˚i. 311˚j.
Five pieces, A, B, C, D and E, of a jigsaw puzzle are shown below. Which four of the given angles will fit together around a point, leaving no gap? 48° A 102° B m C 45° 82° D 165°
In each of the questions below calculate the size of the angles labelled with letters, giving a reason for your answer each time.a. Name the quadrilateral below. Justify your answer.b. Calculate each of the angles d, e, f and g. Justify your answers. 125° dº log e
In the diagram below AC and DE are parallel lines. Prove that triangle BCD is isosceles. m 7 4x+30 B 2x° 130° D
Construct a parallelogram ABCD of side lengths 8 cm and 4cm with internal angles of 108° and 72°. • Draw a line AB, 8 cm long.• Place the protractor on B and mark off an internal angle of 108°.• Draw a line from B through the point marked and measure off 4cm. Label this new point
A kite is shown. a. Which of the following properties are true for a kite?• It has one pair of equal opposite angles.• It has two pairs of equal opposite angles. l Its diagonals intersect at right angles.• It has two pairs of adjacent sides of the same length.• It has one line of
Shapes that tessellate fit together without leaving any gaps.For example, congruent squares and parallelograms tessellate with themselves as shown:Does the kite shown below tessellate with other congruent kites? Justify your answer.
The diagram below shows tessellating parallelograms. a. What is the sum of x+y? Justify your answer.b. Using the diagram above as an aid, prove why any congruent parallelograms will tessellate fully. A xº E 10 F B K L H D
Construct a square of side length 6cm.• Draw a line 6 cm long.• Place a set square at the end of the line, making sure that one of the perpendicular sides rests on the line drawn like this:• Repeat this for the remaining two sides. 6cm 6cm
a. Describe five properties of a parallelogram.b. Calculate the size of the angles d, e and f in the parallelogram below. Justify your answers using some of the properties you identified in part (a). 75° do
The diagram below shows an isosceles trapezium ABCD inside a parallelogram ABED. Angle CDE is 25°.Calculate the size of each of the angles of the trapezium. Justify your answers fully. A B C 25% D E
Calculate the missing number in each mixed number addition below. a.b.c. 1+2=3
The following right-angled triangle has an area of 5h+20 units2. Write down the length of the side marked x. h+4 X
Two rectangles are shown below: a. Write an expression for the area of each rectangle.Rectangle A: 4a × 3a = 12a2 Rectangle B: 4a × 1 = 4ab. Write an expression for the total area of the two rectangles. 12a2 + 4a За 4a A 1 4a B
Write an expression for the area of the following rectangles: a. b. c. d. E+D 4
The following rectangle has dimensions as shown. a. Write a simplified expression for the perimeter of the rectangle. b. A small square of side length p is removed from one corner of the rectangle as shown below.(i) What is the perimeter of the small square? (ii) Choose one of the
The diagram below shows three containers, each with different numbers of red, blue and yellow sweets. a. A blue sweet costs b cents, a red sweet costs r cents and a yellow sweet costs y cents.(i) Write the cost of the sweets in each trayas an algebraic expression. (ii) Write an expression
Write an expression for the area of the following composite shapes. a.b. 8 -j+3- -2j-3- 3
For each of the shapes below write a simplified expression for its perimeter. a.b.c.d.e. 1 W
Six cards are shown below, each with a different algebraic expression. a. Arrange the cards into groups of like terms.b. Simplify each group.c. Write a simplified expression for the sum of all six cards. y 000000 3 3x² 2y 2
The following rectangle has a perimeter of 8x + 16 units. Write an expression for the area of the rectangle. Show your working clearly. 10
Expand the following brackets. a. b. 4(5a-2) 20a-8
A child’s construction set has a number of straight pieces of various lengths as shown. Seven of the eight rods can be arranged to form an equilateral triangle.a. (i) Which seven rods are they? (ii) Show how they can be arranged to make the triangle.b. Write an expression for the length
Work out the answer to the following calculations. Show your working clearly and simplify your answers where possible. a.b.c.d.e.f. 2 5 6
The composite C-shape below has dimensions as shown: a. Which of the following rectangles A, B or C could fit in the gap to transform the C-shape into a rectangle? Justify your answer(s).b. Will your solution(s) in part (a) above definitely fit in the gap of the C-shape? Give convincing
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