Rates — Paper 4 · IGCSE Mathematics - International 0607

10 (a) Sam walks for 30 minutes at 4 km/h and then runs 5 km in 25 minutes. Calculate his average speed. Give your answer in km/h. … km/h [3] (b) (i) Tami walks for 10 minutes at x km/h and then runs y kilometres in z minutes. Find her average speed in terms of x, y and z. Give your answer in km/h, in its simplest form. … km/h [3] (ii) When Tami walks for 10 minutes at 3 km/h and then runs for 20 minutes, her average speed is 11 km/h. Find the distance Tami runs. … km [2] (c) Urs walks for t minutes at 3 km/h and then runs for (t + 10 ) minutes at 7 km/h. 5t + 35 (i) Show that his average speed is km/h. t + 5 [3] (ii) When the average speed is 5 12 km/h, find the value of t. t = … [2]

13 marks

Mark scheme: 10(a) 7.64 or 7.636… 3  30  4 × + 5   60 M2 for   oe 30 25   +   60 60  30 30 25 or M1 for 4 × + 5 or + 60 [60] [60] 10(b)(i) 10 x + 60 y 10( x + 6 y ) 3  10  or x × + y   10 + z 10 + z 60 M2 for   oe 10 z   +   60 60  10 or M1 for total distance = x × + y 60 10 z or total time = + [60] [60] 10(b)(ii) 5 2 M1 for correct substitution of x = 3, z = 20 and average speed = 11 in their formula which must contain x, y and z. or B1 for 5.5 oe or 330 seen 10(c)(i) 3t 7(t + 10) M1 + oe 60 60 t t + 10 M1 The two M1s may be seen together in a + correct fraction [60] [60] Correct simplification to A1 dep on M1M1 5t + 35 At least one line of working and no errors seen t + 5 110(c) (ii) 15 2 M1 for (5t + 35) = ( 5 2 ) (t + 5) oe or better

This question in 0607/41 May/June 2018

10 Isabel drives from Geneva to Rome, a distance of 930 km. Her average speed is x km/h. (a) Write down an expression, in terms of x, for the time, in hours, the journey takes. … h [1] (b) She returns from Rome to Geneva along the same route at an average speed of (x + 5) km/h. 1 The journey takes hour less than the journey from Geneva to Rome. 2 (i) Write down an equation, in terms of x, and show that it simplifies to x2 + 5x – 9300 = 0. [3] (ii) Solve this equation. Give your answers correct to 1 decimal place. x = … or x = … [3] (iii) Find the time taken for the journey from Rome to Geneva in hours and minutes. … h … min [2]

9 marks

Mark scheme: 10(a) 930 1 x 10(b)(i) 930 930 M1 – soi x x+ 5 1860(x + 5) – 1860x = x(x + 5) oe M1 FT dep on 1st M1with 12 as separate term in equation Completion to x2 + 5x – 9300 = 0 A1 with no errors 10(b)(ii) 94[.0] or 93.96 to 93.97 3 2 5 ± 5 − 4(1)( − 9300) –99[.0] or –98.97 to –98.96 M1 for 2 or sketch of parabola B1 for 1 correct 10(b)(iii) 9 [h] 24 or 23 to 24 [min] 2 M1 for 930 ÷ (their 94 + 5) oe If 0 scored SC1 for 9 h 53min to 9 h 54 min

This question in 0607/43 May/June 2018

7 The length of the Jinghu high speed railway from Beijing to Shanghai is 1318 km. (a) A train travels at an average speed of 252 km/h. This train leaves Beijing at 12 49. The local time in Beijing is the same as the local time in Shanghai. Find the time, correct to the nearest minute, that this train arrives in Shanghai. … [4] (b) On the journey this train passes over a bridge of length 6772 m at 252 km/h. The train is 401 m long. (i) Change 252 kilometres per hour to metres per second. … m/s [2] (ii) Calculate the time, in seconds, for the train to completely cross the bridge. … s [2]

8 marks

Mark scheme: 7(a) 18 03 4 M1 for 1318 ÷ 252 A1 for 5.23 or 5.230... M1 for converting their time in hours to hours and minutes 7(b)(i) 70 2 1000 M1 for 252 × oe 60 × 60 7(b)(ii) 102 s or 102.4 to 102.5 2 FT 7173 ÷ their 70 M1 for (6772 + 401) ÷ their 70

This question in 0607/42 Oct/Nov 2018

4 NOT TO SCALE 3 cm l cm The diagram shows a solid made from a cylinder and two hemispheres. The radius of the cylinder and each hemisphere is 3 cm. The total volume of the solid is 144r cm3. (a) The length of the cylinder is l cm. Find the value of l. l = … [3] (b) The solid is made of steel. 1 cm3 of steel has a mass of 7.8 g. Calculate the mass of the solid. Give your answer in kilograms. … kg [2] (c) The solid is melted down and made into 20 cubes each of side length 2.8 cm. Calculate the volume of steel not used for the cubes as a percentage of the 144r cm3. … % [3] (d) A solid that is mathematically similar to the original solid has a volume of 18r cm3. Find the radius of the new cylinder. … cm [3]

11 marks

Mark scheme: 4(a) 12 cao final answer 3 B2 for 11.98 to 12.02 2 2 3 or M1 for π × 3 × l + 2 × × π × 3 [ = 144π] 3 oe 4(b) 3.53 or 3.528 to 3.529... 2 M1 for 144 π × 8.7 soi by figs 353 or 3528 to 3529 4(c) 2.95 or 2.96 or 2.950 to 2.963... 3 144 π − 20 × 8.2 3 M2 for [× 100 ] 144 π 20 × 8.2 3 or × 100 oe 144 π 3 20 × 8.2 3 or M1 for 144 π − 20 × 8.2 or oe 144 π 4(d) 1.5 oe cao final answer 3 B2 for 1.498 to 1.502 18π or M2 for 3 × 3 oe 144 π 18π 144 π or M1 for 3 or 3 oe or better 144 π 18π  3 3 144π or for = oe    x  18π

This question in 0607/42 May/June 2019

9 (a) Lionel runs 10.6 km in 94 minutes. Calculate his average speed in km/h. … km/h [2] (b) Monika walks 2 km at a speed of 4 km/h and then 3 km at a speed of 3 km/h. Calculate Monika’s overall average speed. … km/h [3] (c) A train is travelling at v metres per second. Find an expression, in terms of v, for the speed of the train in kilometres per hour. Give your answer in its simplest form. … km/h [2] (d) (i) A car travels 50 km at x km/h and then 80 km at (x + 10) km/h. Find an expression, in terms of x, for the total time taken, T hours. Give your answer as a single fraction, in its simplest form. T = … h [3] (ii) When T = 2 , show that x 2 - 55x - 250 = 0 . [2] (iii) When T = 2 , find the value of x. x = … [3]

15 marks

Mark scheme: 9(a) 6.77 or 6.765 to 6.766 2 M1 for 10.6 ÷ 94 or 94 ÷ 60 9(b) 1 3 2 + 3 3.33 or 3.333... or 3 M2 for 3 2 3 + 4 3 2 3 or M1 for oe or oe 4 3 9(c) 18 v 3 2 M1 for × (60 × 60) oe or for ÷ 1000 or 3.6v or v 5 35 9(d)(i) 130 x + 500 130 x + 500 3 50 80 or M1 for + x ( x + 10) x 2 + 10 x x x + 10 B1 for common denominator x(x + 10) oe 9(d)(ii) 130 x + 500 = 2 x ( x + 10) oe M1 i.e. fraction with linear numerator and quadratic denominator removed correctly 130 x + 500 = 2 x 2 + 20 x A1 i.e. equation with four terms 2 no errors or omissions leading to x − 55 x − 250 = 0 9(d)(iii) 59.2 or 59.22... only 3 M2 for correct graph of quadratic showing positive root −−( 55) ± ( − 55) 2 − 4(1)( − 250) or for oe 2(1) or M1 for appropriate quadratic graph or for ( −55) 2 − 4(1)( −250) oe −−( 55) or for oe in correct formula 2(1)

This question in 0607/41 Oct/Nov 2019

8 North B NOT TO SCALE 17 km 142° C North 4 km A Rani sails in a boat race around a triangular course. She sails from A to B to C and then directly back to A. B is due north of C. (a) Find the bearing Rani sails on from C to A. … [1] (b) Show that AB = 20.3 km, correct to 1 decimal place. [3] (c) Calculate the bearing of B from A. … [3] (d) Rani starts the race at 08 57 and returns to A at 12 33. Calculate the average speed of her boat in km/h. … km/h [3]

10 marks

Mark scheme: 8(a) 218 1 8(b) 42 + 172 – 2 × 4 × 17 × cos142 M2 M1 for implicit cosine rule 20.30… A1 8(c) 007 or 006.92 to 006.98 3 4sin142 M2 for sin B = oe 20.3 4 20.3 or M1 for = oe sin B sin142 OR 17sin142 M2 for sin A= oe 20.3 17 20.3 or M1 for = oe sin A sin142 8(d) 11.5 or 11.47… 3 B1 for 3 h 36 min or 3.6 h seen 4 + 17 + 20.3 M1 for their 3.6

This question in 0607/42 Feb/March 2021

7 Roisin drives 250 km. She drives the first 200 km at an average speed of x km/h. (a) Write down an expression for the time, in hours, it takes to drive the 200 km. … h [1] (b) For the remainder of the journey, Roisin is in heavy traffic and her average speed is 40 km/h less than for the first 200 km. 1 The total time for the journey is 3 hours. 2 Show that 7x 2 - 780 x + 16000 = 0 . [4] (c) Solve the equation 7x 2 - 780x + 16000 = 0 to find the time taken to travel the first 200 km. Give your answer in hours and minutes correct to the nearest minute. … h … min [5]

10 marks

Mark scheme: 7(a) 200 1 x 7(b) 200 50 7 M1 + = oe x x − 40 2 400(x – 40) +100x = 7x(x – 40) oe M2 FT only an equation of the correct form with equivalent difficulty 200 ( x − 40 ) + 50 x M1FT for x ( x − 40 ) or better Completion to 7x2 – 780x + 16 000 = 0 A1 With at least one intermediate step with no errors or omissions 7(c) 2 [h] 22 [min] 5 B4 for 2.37 or 2.371 to 2.372 or 2h 22 to 22.3... min or 142 to 142.3... or B3 for 27.1 or 27.10 to 27.11 and 84.3 or 84.3... , or M2 for 2 −−( 780 ) ± ( −780 ) − 4 × 7 × 16000 2 × 7 or sketch of parabola (positive x2 ) with two positive zeros or M1 for ( − 780 ) 2 − 4 × 7 × 16000 −−( 780 ) ± p or 2 × 7 and M1 for 200 ÷ their solution from their quadratic if > 40

This question in 0607/42 May/June 2021

11 NOT TO SCALE 2.1 m 110° 110° 0.9 m The diagram shows the symmetrical cross-section of a ditch containing water. The angle between the base and each side of the ditch is 110°. The width of the base is 0.9 m and the depth of the water is 2.1 m. The ditch is 100 m long. (a) Calculate the volume of water in the ditch. … m3 [4] (b) On a different day, the ditch contains 300 m 3of water. Water is pumped out of the ditch at a rate of 4.2 litres per second. Calculate the time taken to empty the ditch completely. Give your answer in hours and minutes, correct to the nearest minute. … h … min [4]

8 marks

Mark scheme: 11(a) 349 or 350 or 349.4 to 349.5… 4 M3 for [0.9  2.1  2  12 (2.1  2.1tan 20)] × 100 oe 1 or (0.9  (2  2.1tan 20  0.9))  2.1 × 2 100 oe or M2 for area of cross section 0.9  2.1  2[ 12 (2.1  2.1tan 20)] oe 1 or (0.9  (2  2.1tan 20  0.9))  2.1 2 oe or M1 for tan 20 x 2.1 If 0 scored SC1 for triangle marked/drawn with 20 or 70 11(b) 19 h 50 (or 51) min 4 B3 for 19.8 or 19.84… 300  1000 or M2 for oe 4.2  60  60 or M1 for 300  1000 or 4.2  60  60 or for their volume divided by their rate

This question in 0607/41 May/June 2022

8 North NOT TO B 120° SCALE North 65 km C 55° A The diagram shows the route of a ship between three ports, A, B and C. The bearing of B from A is 055° and the bearing of C from B is 120°. BC = 65 km . The ship takes 7 hours to sail from A to B. It sails at a speed of 20 km/h. (a) Find the distance AB. … km [1] (b) Show that angle ABC = 115° . [1] (c) (i) Calculate the distance CA. … km [3] (ii) Calculate the bearing of A from C. … [4] (d) The ship takes 3.6 hours to sail from B to C. It then sails from C to A at a speed of 21.5 km/h. Find the average speed for the complete journey from A to B to C and back to A. … km/h [3]

12 marks

Mark scheme: 8(a) 140 1 8(b) 360 – (120 + 125) or 60 + 55 1 or 180 + 55 – 120 8(c)(i) 178 or 177.5... 3 M2 for ((their140) 2  65 2  2  ( their140)  65  cos115) OR M1 for (their 140)2 + 652 – 2 × (their 140) × 65 × cos115 8(c)(ii) 254 or 255 or 254.3 to 254.5... 4 their140sin115 M2 for sin[C ]  oe their178 sin[ C ] sin115 or M1 for  oe their140 their178 A1 for 45.18 to 45.63 M1 for 360 – 60 – their C oe calculated as answer 8(d) 20.3 or 20.26 to 20.31... 3 their140  65  their178 M2 for their178 7  3.6  21.5 their178 or M1 for 21.5 totaldistance or for clear indication of totaltime

This question in 0607/43 May/June 2022

11 A tank has a capacity of 400 litres. Water from Tap A flows at x litres per minute. Water from Tap B flows at 2 litres per minute less than the water from tap A. (a) Write down an expression in terms of x for the time, in minutes, for tap A to fill the tank. … [1] (b) Tap B takes 10 minutes longer to fill the tank than tap A. Write down an equation in terms of x and show that it simplifies to x 2 - 2x - 80 = 0 . [4] (c) Solve x 2 - 2x - 80 = 0 and find the time it takes to fill the tank when both taps are turned on. Give your answer in minutes and seconds, correct to the nearest second. … minutes … seconds [4]

9 marks

Mark scheme: 11(a) 400 1 x 11(b) 400 400 M1 – = 10 oe x  2 x 400x – 400(x – 2) = 10x(x – 2) M1 FT clearing fractions. Dep on equation in the right form –400x + 800 or 400x – 800 M1 FT Expansion of both brackets of correct form and 10x2– 20x Completion to x² – 2x – 80 = 0 A1 with no errors or omissions 11(c) 22 min 13 sec 4 B2 for x = 10 isw other value of x or M1 for (x – 10)(x + 8) ( 2)  ( 2) 2  4(1)( 80) or 2  1 or sketch of parabola with one positive zero and one negative zero 400 M1 for their 10  their10  2

This question in 0607/43 May/June 2022

10 (a) A machine lays a pipe of length 2.5 km in 18 hours. The machine always works at the same rate. Calculate the time it takes to lay a pipe of length 4 km. … hours [2] (b) t varies inversely as the square root of x. x varies directly as the square of y. When x = 4, t = 3 . When y = 4, x = 81. ty = h Find the value of h. h = … [5]

7 marks

Mark scheme: 10(a) 28.8 2 4 M1 for  18 oe 2.5 10(b) 8 5 6 16 x oe B4 for ty =  oe 3 x 81 2 2 36 16 x or t y =  oe x 81 6 81 2 B3 for t = oe and x = y oe x 16 6 81 2 or B2 for t = oe or x = y x 16 oe k or M1 for t = oe or x = ky2 oe x

This question in 0607/43 Oct/Nov 2022

2 A triathlon race consists of three parts: • a 1500 m swim • a 40 km bike ride • a 10 km run. (a) John swims the 1500 m in 25 minutes. Find his average speed, in km/h, for this swim. … km/h [2] (b) John completes the 40 km bike ride at an average speed of 32 km/h. Find the time, in minutes, for John to complete this bike ride. … min [2] (c) John completes the whole race at an average speed of 20.6 km/h. Find the average speed, in km/h, for John to complete his 10 km run. … km/h [3]

7 marks

Mark scheme: 2(a) 3.6 oe 2 1500 1.5 25 M1 for or or figs15  25 25 60 2(b) 75 2 40 M1 for 32 2(c) 12 3 M1 for 51.5  20.6 oe M1 for 10   their150  25  their (b )  [ 60] oe

This question in 0607/43 May/June 2023

6 (a) (i) Kayla walks from A to B on a bearing of 105°. She then walks back to A. Calculate the bearing Kayla walks from B to A. … [2] (ii) The distance from A to B is 1.5 km. (a) It takes Kayla 24 minutes to walk from A to B. Calculate her average speed in km/h. … km/h [2] (b) Kayla has a map with a scale of 1 : 25 000 showing A and B. Work out the length of AB on the map. Give your answer in centimetres. … cm [2] (b) A train is 770 m long. The train takes 2 minutes and 36 seconds to travel completely through a tunnel. Its speed through the tunnel is 120 km/h. Work out the length of the tunnel. Give your answer in metres. … m [4]

10 marks

Mark scheme: 6(a)(i) 285 2 M1 for 360 – (180 – 105) oe or a sketch with correct indication of 75 or 105 at B 6(a)(ii)(a) 3.75 oe 2 1.5   60  M1 for 24 6(a)(ii)(b) 6 2 1.5  1000  100  M1 for oe 25000 6(b) 4430 4 B3 for 5200 OR  36   2 +  36  60  M1 for 2 + or ( 2  60 ) + 36 or 60 60 36 M1 for their 2  120 [ ] 60 or their ( ( 2  60 ) + 36 )  120 [ ] 36 120  1000 M1 for their 2  60 60 120  1000 or their ( ( 2  60 ) + 36 )  60  60

This question in 0607/42 Oct/Nov 2023

1 (a) In 1911 the men’s world record for the triple jump was 15.52 m. In 2021 the record was 18.29 m. Find 15.52 m as a percentage of 18.29 m. … % [1] (b) In 2021 the women’s world record for running 800 m was 1 minute 53 seconds. Find the average speed for this run in m/s. … m/s [2] (c) In 2021 the men’s world record speed for running 100 m was 37.58 km/h. Find the time taken, in seconds, for this run. … s [3]

6 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 84.9 or 84.85 to 84.86 1 1(b) 7.08 or 7.079 to 7.08 2 M1 for distance divided by time. or B1 for 113 1(c) 9.58 or 9.579 to 9.580 3 M1 for distance divided by speed M1 for ÷ 1000 and × 60  60

This question in 0607/43 Oct/Nov 2023

2 Asif, Basheera and Chelsea make baskets. (a) The selling price of a basket increases by 8%. The new selling price is $4.86 . Find the original selling price of a basket. $ … [2] (b) Asif earns $4.70 per hour plus $1.21 for each basket he makes. Each week he works 8 hours a day for 5 days. Each day Asif makes 18 baskets. Calculate the total amount Asif earns in one week. $ … [3] (c) One day Basheera and Chelsea make a total of 36 baskets. They each work for 8 hours. Basheera takes x minutes to make a basket. Basheera takes 6 minutes longer than Chelsea to make a basket. (i) Write down an expression in terms of x for the number of baskets Chelsea makes. … [1] (ii) Write down an equation in terms of x and show that it simplifies to 3x 2 - 98x + 240 = 0 . [3] (iii) Solve the equation 3x 2 - 98 x + 240 = 0 . x = … or … [2] (iv) Find the number of baskets Chelsea makes. … [2]

13 marks

Mark scheme: 2(a) 4.5[0] 2  8  M1 for A   1 +  = 4.86 oe  100  2(b) 296.9[0] 3 M1 for 8 [× 5] × 4.7[0] oe M1 for [5 ×] 18 × 1.21 oe 2(c)(i) 480 1 oe x − 6 2(c)(ii) 480 480 M1 480 + = 36 oe Allow + their (i) = 36 x x − 6 x 480(x – 6) + 480x = 36x(x – 6) oe M1 Clearing fractions correctly (Dep on two fractions with different algebraic denominators) Completion to A1 3x2 – 98x + 240 [= 0] with no errors 2(c)(iii) 8 2 M1 for suitable sketch of parabola with 2 30, oe positive solutions 3 or (3x – 8)(x – 30) = 0 −−( 98)  ( −98) 2 − 4(3)(240) or 2  3 2(c)(iv) 20 2 480 M1 for their 30 − 6

This question in 0607/42 Feb/March 2024