A sheet of rectangular paper has length $l$ cm and breadth $b$ cm. If identical squares of side 3 cm are to be cut out from this rectangular paper, there will not be any leftovers. If we rearrange some of these identical rectangular papers to form a square, the smallest possible square formed will have side 30 cm. Given that both $l$ and $b$ are greater than 3 cm and $l > b$, find the values of $l$ and $b$. Answer $l =$ ... $b =$ ... The time taken by Shayne to swim a lap is 40% more than the time taken by Kai. Is it true that the time taken by Kai was 60% of the time by Shayne? Explain your answers with appropriate workings. A sweet drink is made by mixing concentrated syrup and water together. The ratio of syrup to water in a 200 ml sweet drink is 9 : 11. Find the volume of water that must be added to the 200 ml sweet drink so that the percentage concentration of syrup will be reduced by 10%. Answer ... ml

1. Represent Shayne's time in terms of Kai's time

Let the time taken by Kai be $x$ seconds.

Shayne's time is $40\%$ more than Kai's time, so
$\text{Shayne's time}=140\% \times x=1.4x.$

2. Find Kai's time as a percentage of Shayne's time and conclude

Kai's time as a percentage of Shayne's time is
$\frac{x}{1.4x}\times 100\%=71.428\ldots\%.$

So Kai's time was about $71.4\%$ of Shayne's time, not $60\%$.

1. Find the volume of syrup in the original 200 ml drink

Ratio of syrup to water $=9:11$

Total parts $=9+11=20$

Volume of syrup
$=\frac{9}{20}\times 200=90\text{ ml}$

2. Find the new percentage concentration of syrup after a 10% reduction

Original percentage concentration of syrup
$=\frac{90}{200}\times 100\%=45\%$

Reduced by $10\%$:
$45\%\times 90\%=40.5\%$

3. Form an equation using the unchanged syrup volume after adding water

Let the volume of water added be $x$ ml.

New total volume $=(200+x)$ ml.

The volume of syrup remains $90$ ml, so
$\frac{90}{200+x}\times 100\%=40.5\%$

4. Solve for the volume of water added

$\frac{90}{200+x}=0.405$

$200+x=\frac{90}{0.405}=222.222\ldots$

$x=222.222\ldots-200=22.222\ldots$

Therefore, the volume of water added is
$22.2\text{ ml}$
correct to 3 significant figures.