Grade 12 Maths · Euclidean Geometry

Use a ratio, rate or proportion

Use ratios, rates and proportion to share amounts or find missing values (for example share R110 in the ratio 4 : 7, giving R40 and R70).

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Ratio sharing the total - Corbettmathscorbettmaths

Direct and Indirect ProportionKevinmathscience

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Question 10.2 · May/June 2025 · Paper 1 · National (DBE) · 6 marks

10.2 A game at a fun park requires a player to roll a six-sided dice and pick a card from a deck of 52 cards. A player wins if an odd number appears on the uppermost face of the dice and the player also draws a picture card from the deck. A deck of cards has 4 suites (hearts, diamonds, spades and clubs). There are 4 picture cards (king, queen, jack and ace) in every suite. A player pays R10 to play a game and in an hour, 260 people can each play one game. If the owner wants to make a 70% profit per hour, calculate the maximum amount that the owner must pay out to each winner.(6)

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3. Question 10.2.3 · May/June 2025 · Paper 2 · National (DBE) · 6 marks
10.2 In the diagram, O is the centre of the circle. Points D and B lie on the circle. Points A and C lie outside the circle such that side AC of $Δ A D C$ passes through B. F is a point on BC such that $FO ∥ B D$ . $D R = RB$ and RO is drawn.
10.2.3 It is further given that $D C = 19, 2$ units, $B D = 12$ units and $\frac{RO}{R D} = \frac{3}{4}$ . Prove, with reasons, that $BF = \frac{75}{16}$ .(6)
Show the memo (3 ticks)Memo
1. ✓
  Find lengths of RD and ROGiven $D R = RB$ , so R is the midpoint of chord BD.
  $R D = \frac{1}{2} \times 12 = 6$ units.
  $\frac{RO}{R D} = \frac{3}{4}$
  $RO = \frac{3}{4} (6) = 4, 5$ units
  2 marks · Use a ratio, rate or proportion
2. ✓
  Calculate DO using PythagorasThe line from the centre (O) to the midpoint (R) of a chord (BD) is perpendicular to the chord.
  $∴ OR ⊥ B D$
  
  In $Δ D RO$ :
  $D O^{2} = R D^{2} + R O^{2}$ [Pythagoras]
  $D O = 6^{2} + 4, 5^{2}$
  $D O = 7, 5$ units
  2 marks · Use the theorem of Pythagoras
3. ✓
  Use proportionality theorem in the large triangle to find BFIn $Δ BC D$ , $FO ∥ B D$ (given):
  $\frac{BF}{BC} = \frac{D O}{D C}$ [line $∥$ one side of $Δ$ OR prop theorem]
  
  From 10.2.2, we know $BC = B D = 12$ units.
  $\frac{BF}{12} = \frac{7 , 5}{19 , 2}$
  $BF = \frac{7 , 5 \times 12}{19 , 2}$
  $BF = \frac{75}{16}$ units
  2 marks · Use the proportionality theorem
4. Question 9.2 · March 2021 · Provincial · 6 marks
The diagram is the top view design of a new railway system. AF || BE and AC || GD. $\frac{A B}{BC} = \frac{4}{7}$ and $\frac{A G}{A F} = \frac{9}{17}$ .
9.2 If the straight line distance of the track from F to C is 374 km and it takes 50 hours to build one km of the track, determine the number of hours it will take to build the section from E to D.(6)
Show the memo (6 ticks)Memo
1. ✓
  $FE = 4 p$ and $EC = 7 p$ Let $FE = 4 p$ and $EC = 7 p$ (since $\frac{FE}{FC} = \frac{4}{11}$ )
  1 mark · Use a ratio, rate or proportion
2. ✓
  $p = 34$ $11 p = 374 ⟹ p = 34$
  $FE = 4 (34) = 136$ km
  1 mark · Solve a linear equation
3. ✓
  $F D = 8 m$ and $D C = 9 m$ Let $F D = 8 m$ and $D C = 9 m$ (since $\frac{C D}{D F} = \frac{9}{8}$ )
  1 mark · Use a ratio, rate or proportion
4. ✓
  $m = 22$ $17 m = 374 ⟹ m = 22$
  $C D = 9 (22) = 198$ km
  1 mark · Solve a linear equation
5. ✓
  $E D = 40$ km $E D = 374 - 4 p - 9 m = 374 - 136 - 198 = 40$ km
  1 mark · Calculate with integers
6. ✓
  Answer: 2 000 hoursTime $= 40 \times 50 = 2000$ hours
  1 mark · Use a ratio, rate or proportion
5. Question 2.1 · September 2024 · Provincial · 3 marks
2.1 On a trip to America, Charity booked into a hotel for five days. The exchange rate at that time was $\$ 1 = R18{,}73 $. T h ecos t o f t h e h o t e lp er d a y w a s$ \ $104$ . How much did she spend, in rands, for five days?(3)
Show the memo (2 ticks)Memo
1. ✓
  $R 1947, 92$ (double tick) $\$ 104 = x$
  $x = 104 \times 18, 73$
  $x = R 1947, 92$ per day
  2 marks · Convert currency using exchange rates
2. ✓
  Calculate total cost for 5 days $Total = R 1947, 92 \times 5$
  $= R 9739, 60$
  1 mark · Use a ratio, rate or proportion
6. Question 4.1 · November 2024 · Paper 1 · Provincial · 3 marks
4.1 Mark has booked himself a hotel room for three nights during his visit to New York. The exchange rate during his visit was $\$ 1 = R~9{,}10 $. T h ecos tp er ni g h t o f t h e h o t e lw a s$ \ $450$ . How much did the three nights cost him in rands?(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate total dollar cost $3 \times \$ 450 = \ $1350$
  1 mark · Use a ratio, rate or proportion
2. ✓
  Convert to Rands $\$ 1 350 \times R9{,}10$
  1 mark · Convert currency using exchange rates
3. ✓
  Final Answer $= R 12285$
  1 mark · Convert currency using exchange rates

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