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Ratios and Proportions

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Question 121Question →

Match each military work scenario with the correct time it would take to complete the task. Assume all personnel work at the same constant rate.

Click a left item, then click its matching right item

Items

2 mechanics take 6 hours to repair an engine. How many hours will it take 4 mechanics?
3 medics take 8 hours to inventory supplies. How many hours will it take 6 medics?
4 logisticians take 5 hours to load a transport. How many hours will it take 10 logisticians?
5 engineers take 6 hours to clear a path. How many hours will it take 6 engineers?

Matches

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Answer

The 4 mechanics will take 3 hours, the 6 medics will take 4 hours, the 10 logisticians will take 2 hours, and the 6 engineers will take 5 hours.
Each scenario is an inverse proportion problem where the product of workers and time is a constant (total work). To find the new time, multiply the original workers by the original hours, then divide by the new number of workers.

Step-by-Step Solution

1
Calculate the constant total work for each scenario.
Scenario 1: 12 hours. Scenario 2: 24 hours. Scenario 3: 20 hours. Scenario 4: 30 hours.
In inverse proportions, the product of workers and time is a constant (representing the total work required).
2
Divide the total work by the new number of workers.
12/4 = 3 hours; 24/6 = 4 hours; 20/10 = 2 hours; 30/6 = 5 hours.
Since the total work remains constant, dividing it by the available workforce yields the new time.

Key Concept

Inverse Proportions
Question 122Question →

A military supply depot distributes a shipment of Meals Ready-to-Eat (MREs) among three battalions: Alpha, Bravo, and Charlie. Battalion Alpha receives 13\frac{1}{3} of the total shipment. The remaining MREs are then divided between Battalion Bravo and Battalion Charlie in a ratio of 3:53 : 5 respectively. If Battalion Charlie receives exactly 1,2001,200 more MREs than Battalion Bravo, what is the total number of MREs in the entire shipment?

Show answer & explanation

Answer: 7200

Answer

The total number of MREs in the entire shipment is 7,200.
The correct total shipment is found by reasoning backward from the difference. Bravo and Charlie split the remaining 2/3 of the shipment in a 3:5 ratio, meaning there are 8 total parts for that remainder. Charlie receives 2 more parts than Bravo (5−3=25 - 3 = 2). Since these 2 parts equal 1,200 MREs, 1 part equals 600 MREs. The entire remainder (all 8 parts) is therefore 8×600=4,8008 \times 600 = 4,800 MREs. Since this 4,800 represents the 2/3 of the total shipment left after Alpha's portion, the entire original shipment must be (4,800÷2)×3=7,200(4,800 \div 2) \times 3 = 7,200 MREs.

Step-by-Step Solution

1
Define a variable for the total shipment and calculate the remaining amount after Alpha's share is removed.
Let the total shipment be T. Alpha receives 13T\frac{1}{3}T, so the remaining portion for Bravo and Charlie is T−13T=23TT - \frac{1}{3}T = \frac{2}{3}T.
We must determine exactly what portion of the whole shipment is being divided between the remaining two battalions.
2
Determine the fractional share of the remainder that Bravo and Charlie each receive based on their 3:53:5 ratio.
The ratio 3:53:5 has 3+5=83 + 5 = 8 total parts. Bravo gets 38\frac{3}{8} of the remainder, and Charlie gets 58\frac{5}{8} of the remainder.
Ratios represent parts of a whole; in this step, the 'whole' is the remaining 23T\frac{2}{3}T of the shipment.
3
Express Bravo's and Charlie's individual shares in terms of the total original shipment T.
Bravo's share is 38×23T=14T\frac{3}{8} \times \frac{2}{3}T = \frac{1}{4}T. Charlie's share is 58×23T=512T\frac{5}{8} \times \frac{2}{3}T = \frac{5}{12}T.
Converting their shares back to fractions of the original total allows us to set up an algebraic equation for the entire shipment.
4
Set up an equation using the known difference between Charlie's and Bravo's shares.
Charlie receives 1,200 more than Bravo, so: 512T−14T=1200\frac{5}{12}T - \frac{1}{4}T = 1200.
This translates the word problem's condition into a solvable algebraic equation.
5
Solve the equation for T.
Convert 14T\frac{1}{4}T to 312T\frac{3}{12}T. The equation becomes 212T=1200\frac{2}{12}T = 1200, which simplifies to 16T=1200\frac{1}{6}T = 1200. Multiplying both sides by 6 yields T=7200T = 7200.
Isolating the variable gives the final total number of MREs.

Key Concept

Dividing a remaining quantity into ratio parts and utilizing the algebraic difference between parts to find an original whole.
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