Match each inverse proportion scenario to its correct calculated value.

Base rations feed $500$ troops for $30\text{ days}$ . How long will the same rations last for $750$ troops? $20\text{ days}$
A transport plane traveling at $400\text{ mph}$ completes a route in $6\text{ hours}$ . How long will the flight take at $600\text{ mph}$ ? $4\text{ hours}$
A detail of $12$ sailors paints a deck in $8\text{ hours}$ . How long will it take $16$ sailors working at the same rate? $6\text{ hours}$
A $60$ -tooth gear spins at $120\text{ RPM}$ . What is the rotational speed of a connected $45$ -tooth gear? $160\text{ RPM}$

Answer

Each scenario represents an inverse proportion and must be matched with the value that maintains the constant product of the initial conditions.

Each match is mathematically determined by setting up an inverse proportion equation where the product of the initial states equals the product of the final states (

x_1 \cdot y_1 = x_2 \cdot y_2

). Dividing the initial product by the new given value provides the correct corresponding answer.

Step-by-Step Solution

Identify that each scenario represents an inverse relationship.

As one quantity increases (troops, speed, workers, gear teeth), the other quantity must decrease.

This determines that the product of the variables is constant (

x_1 \cdot y_1 = x_2 \cdot y_2

), rather than the ratio.

Calculate the constant product (

k

) for the initial conditions in each scenario.

The constants are

15,000

troop-days,

2,400\text{ miles}

96\text{ man-hours}

, and

7,200\text{ teeth-RPM}

The constant represents the total amount of work, distance, resources, or mechanical transfer that must remain unchanged.

Divide the constant product by the new given variable to find the unknown required value.

The resulting values are

20\text{ days}

4\text{ hours}

6\text{ hours}

, and

160\text{ RPM}

Solving the equation

x_1 \cdot y_1 = x_2 \cdot y_2

for the unknown variable

y_2

requires dividing the constant by

x_2

Key Concept

Inverse Proportions

Question

Answer

Step-by-Step Solution

Key Concept

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