Proportional Reasoning-2 - Inverse Proportions

If $24$ workers can build a wall in $15$ days, how many days will $9$ workers take to build the same wall?

Let the number of workers be $x$ and the number of days be $y$. Since more workers will take fewer days, this is a case of inverse proportion. Given: $x_1 = 24$, $y_1 = 15$, and $x_2 = 9$. We need to find $y_2$. Using the formula: $x_1 y_1 = x_2 y_2$ $24 \times 15 = 9 \times y_2$ $360 = 9 \times y_2$ $y_2 = \frac{360}{9}$ $y_2 = 40$

Because the amount of work is constant, the product of workers and days stays the same ($24 \times 15 = 360$). To find the new number of days, divide the total work units ($360$) by the new number of workers ($9$), resulting in $40$ days.

A box of sweets is divided among $28$ children, and they get $5$ sweets each. How many sweets would each get if the number of children is reduced by $8$?

Let the number of children be $x$ and the sweets per child be $y$. Total children initially $x_1 = 28$. Sweets per child $y_1 = 5$. New number of children $x_2$ is: $\begin{array}{r} 28 \\ -8 \\ \hline 20 \end{array}$ So, $x_2 = 20$. Since the total number of sweets is constant, $x_1 y_1 = x_2 y_2$. $28 \times 5 = 20 \times y_2$ $140 = 20 \times y_2$ $y_2 = \frac{140}{20}$ $y_2 = 7$

First, calculate the new number of children ($28 - 8 = 20$). In inverse proportion, as the number of children decreases, the number of sweets per child increases. The product $28 \times 5 = 140$ represents the total sweets. Dividing $140$ by $20$ children gives $7$ sweets each.

A car travels at a speed of $60 \text{ km/h}$ to cover a distance in $3$ hours. How much time will it take to cover the same distance at a speed of $80 \text{ km/h}$?

Let speed be $s$ and time be $t$. Speed and time are inversely proportional for a fixed distance. $s_1 = 60, t_1 = 3$ $s_2 = 80, t_2 = ?$ Using $s_1 t_1 = s_2 t_2$: $60 \times 3 = 80 \times t_2$ $180 = 80 \times t_2$ $t_2 = \frac{180}{80}$ $t_2 = 2.25 \text{ hours}$ Converting to hours and minutes: $0.25 \text{ hours} = 0.25 \times 60 = 15 \text{ minutes}$. So, $t_2 = 2 \text{ hours } 15 \text{ minutes}$.

The distance remains constant, so the product of speed and time is constant ($180 \text{ km}$). When the speed increases to $80 \text{ km/h}$, the time taken decreases to $2.25$ hours.