一张矩形纸片的长度为 $l$ cm，宽度为 $b$ cm。 如果从这张矩形纸片上剪下边长为 3 cm 的相同正方形，将不会有任何剩余。 如果我们将若干张这样相同的矩形纸片拼接成一个正方形，所能拼成的最小正方形的边长将为 30 cm。 已知 $l$ 和 $b$ 都大于 3 cm 且 $l > b$，求 $l$ 和 $b$ 的值。 答案 $l =$ ... $b =$ ... Shayne 游完一圈所用的时间比 Kai 多 40%。 Kai 所用的时间是 Shayne 所用时间的 60%，这种说法对吗？ 请通过适当的计算过程解释你的答案。 一种甜饮料是由浓缩糖浆和水混合制成的。 在 200 ml 的甜饮料中，糖浆与水的比例是 9 : 11。 求必须往这 200 ml 甜饮料中加入多少体积的水，才能使糖浆的百分比浓度降低 10%。 答案 ... 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.