Level 4 — ApplicationArithmetic & Number Systems

Arithmetic & Number Systems

⏱ 60 minutes50 marksprintable — key stays hidden on paper

Level: 4 (Application — novel problems, no hints) Time limit: 60 minutes Total marks: 50

Question 1. (HCF/LCM applications) [10 marks]

Three bells at a temple ring at intervals of $18$ , $24$ , and $30$ seconds respectively. They all ring together at $6{:}00{:}00$ am.

(a) After how many seconds will they next all ring together? [4] (b) How many times (including the start at $6{:}00{:}00$ ) will they ring together during the interval from $6{:}00{:}00$ am up to and including $6{:}30{:}00$ am? [4] (c) Verify the relationship $\text{HCF}\times\text{LCM} = \text{product}$ for the two numbers $24$ and $30$ only, showing all working. [2]

Question 2. (Percentages, profit/loss, discount) [12 marks]

A shopkeeper marks a jacket at $\$ 1200$.

(a) He offers a discount of $15\%$ . Find the selling price after discount. [3] (b) After the discount he still makes a profit of $19\%$ on his cost price. Find his cost price (round to the nearest cent). [4] (c) During a sale he instead sells the jacket at a loss of $8\%$ on the cost price found in (b). What single percentage discount off the marked price of $\$ 1200$ does this loss-price represent? Round to one decimal place. [5]

Question 3. (Fractions, decimals, order of operations) [10 marks]

(a) Evaluate, showing each BODMAS step: $\left[\,\frac{3}{4} + \frac{2}{3}\left(\frac{5}{6} - \frac{1}{2}\right)\right] \div \frac{7}{8}$ [6]

(b) A tank is $\frac{3}{5}$ full. After $18$ litres are removed it is $\frac{2}{5}$ full. Find the total capacity of the tank. [4]

Question 4. (Ratio, unitary method, proportion) [10 marks]

(a) A sum of money is divided among Amy, Ben and Cara in the ratio $4:5:7$ . Cara receives $\$ 126$ more than Amy. Find the total sum. [5]

(b) $8$ workers can build a wall in $15$ days. Working at the same rate, how many workers are needed to build the same wall in $10$ days? State whether this is direct or inverse proportion and justify. [5]

Question 5. (Number systems, place value, absolute value) [8 marks]

(a) Write the number "three crore five lakh forty thousand and seven" in figures using the Indian place value system, then state the digit in the ten-thousands place. [3]

(b) Evaluate $|{-13} + 5| - |7 - 15| + |{-4}|$ . [3]

Answer keyMark scheme & solutions

Question 1

(a) Need LCM(18, 24, 30). Prime factorizations: $18 = 2\cdot3^2$ , $24 = 2^3\cdot3$ , $30 = 2\cdot3\cdot5$ . [1] LCM = $2^3\cdot3^2\cdot5 = 360$ . [2] They next ring together after 360 seconds (= 6 minutes). [1]

(b) From $6{:}00{:}00$ to $6{:}30{:}00$ is $30$ min $= 1800$ s. [1] Ring together at $t = 0, 360, 720, 1080, 1440, 1800$ s. [2] Number of times $= \frac{1800}{360} + 1 = 5 + 1 = \mathbf{6}$ times. [1]

(c) HCF(24,30): $24=2^3\cdot3$ , $30=2\cdot3\cdot5$ , common $=2\cdot3=6$ . LCM(24,30) $=2^3\cdot3\cdot5=120$ . [1] $\text{HCF}\times\text{LCM}=6\times120=720$ ; product $=24\times30=720$ . Equal ✓. [1]

Question 2

(a) Discount $=15\%$ of $1200 = 0.15\times1200 = 180$ . [1] Selling price after discount $= 1200 - 180 = \mathbf{\$ 1020}$. [2]

(b) SP $= 1020 = \text{CP}\times(1+0.19) = 1.19\,\text{CP}$ . [2] CP $= \dfrac{1020}{1.19} = 857.142\ldots \approx \mathbf{\$ 857.14}$. [2]

(c) Loss price $= \text{CP}\times(1-0.08) = 857.142857\times0.92 = 788.5714\ldots \approx \$ 788.57 $. **[2]** As discount off marked price$ 1200 $: discount amount$ = 1200 - 788.57 = 411.43 $. **[1]** Percentage discount$ = \dfrac{411.43}{1200}\times100 = 34.29% \approx \mathbf{34.3%} $. **[2]** (Using exact fractions: loss price$ =\frac{1020}{1.19}\times0.92 = \frac{1020\times0.92}{1.19}=788.571\ldots $;$ %=\frac{1200-788.571}{1200}\times100=34.286%\approx34.3%$.)

Question 3

(a) Innermost bracket: $\frac{5}{6}-\frac{1}{2} = \frac{5}{6}-\frac{3}{6} = \frac{2}{6} = \frac{1}{3}$ . [1] Multiply: $\frac{2}{3}\cdot\frac{1}{3} = \frac{2}{9}$ . [1] Add: $\frac{3}{4}+\frac{2}{9}$ ; LCD $=36$ : $\frac{27}{36}+\frac{8}{36} = \frac{35}{36}$ . [2] Divide by $\frac{7}{8}$ : $\frac{35}{36}\times\frac{8}{7} = \frac{35\cdot8}{36\cdot7} = \frac{280}{252} = \frac{10}{9}$ . [2] Answer: $\dfrac{10}{9} = 1\frac{1}{9}$ .

(b) Drop in level $= \frac{3}{5}-\frac{2}{5} = \frac{1}{5}$ of capacity. [2] $\frac{1}{5}$ of capacity $= 18$ L $\Rightarrow$ capacity $= 18\times5 = \mathbf{90\ \text{litres}}$ . [2]

Question 4

(a) Ratio parts $4:5:7$ . Cara $-$ Amy $= 7-4 = 3$ parts $= \$ 126 $. **[2]**$ 1 $part$ = 126/3 = $42 $. **[1]** Total parts$ = 4+5+7 = 16 $. **[1]** Total sum$ = 16\times42 = \mathbf{$672}$. [1]

(b) Total work $= 8\times15 = 120$ worker-days. [2] For $10$ days: workers $= 120/10 = \mathbf{12}$ workers. [1] This is inverse proportion: fewer days requires more workers (product workers×days is constant). [2]

Question 5

(a) Three crore = $3{,}00{,}00{,}000$ ; five lakh $=5{,}00{,}000$ ; forty thousand $=40{,}000$ ; seven $=7$ . Number $= \mathbf{3{,}05{,}40{,}007}$ . [2] Ten-thousands place digit $= \mathbf{4}$ . [1]

(b) $|-13+5| = |-8| = 8$ ; $|7-15| = |-8| = 8$ ; $|-4| = 4$ . [2] Value $= 8 - 8 + 4 = \mathbf{4}$ . [1]

(c) $|n-2|\le3 \Rightarrow -3\le n-2\le3 \Rightarrow -1\le n\le5$ . [1] Integers: $\mathbf{-1, 0, 1, 2, 3, 4, 5}$ . [1]

[
  {"claim":"LCM of 18,24,30 is 360 and count of coincidences in 1800s is 6","code":"L=ilcm(18,24,30); result=(L==360 and (1800//L+1)==6)"},
  {"claim":"Cost price rounds to 857.14 from SP 1020 at 19% profit","code":"cp=Rational(1020,1)/Rational(119,100); result=(round(float(cp),2)==857.14)"},
  {"claim":"Sale loss-price discount off 1200 is 34.3% (1 dp)","code":"cp=Rational(1020)/Rational(119,100); lossp=cp*Rational(92,100); disc=(Rational(1200)-lossp)/Rational(1200)*100; result=(round(float(disc),1)==34.3)"},
  {"claim":"BODMAS expression equals 10/9","code":"expr=(Rational(3,4)+Rational(2,3)*(Rational(5,6)-Rational(1,2)))/Rational(7,8); result=(expr==Rational(10,9))"},
  {"claim":"Ratio total sum is 672","code":"part=Rational(126,3); result=(16*part==672)"},
  {"claim":"Absolute value expression equals 4","code":"result=(abs(-13+5)-abs(7-15)+abs(-4)==4)"}
]