4 sets of uniforms were issued to each combat soldier and 3 sets of uniforms were issued to each non-combat soldier in an army camp. The ratio of the number of soldiers in the camp to the number of uniforms issued was 7:25. What fraction of the soldiers in the army camp were combat soldiers?

Question

4 sets of uniforms were issued to each combat soldier and 3 sets of uniforms were issued to each non-combat soldier in an army camp. The ratio of the number of soldiers in the camp to the number of uniforms issued was  7:25. What fraction of the soldiers in the army camp were combat soldiers?

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Identify uniforms per soldier type: Identify the total number of uniforms issued per soldier type. Combat soldiers: 4 uniforms each, Non-combat soldiers: 3 uniforms each. Define variables and total: Let x be the number of combat soldiers and y be the number of non-combat soldiers. The total number of soldiers is x + y. The total number of uniforms issued is 4x + 3y. Set up ratio equation: Given the ratio of the number of soldiers to the number of uniforms is 7:25. Set up the equation (x + y) / (4x + 3y) = 7 / 25. Cross-multiply for x and y: Cross-multiply to solve for x and y: 25(x + y) = 7(4x + 3y). Simplify to 25x + 25y = 28x + 21y. Rearrange equation: Rearrange the equation: 25x + 25y - 28x - 21y = 0, which simplifies to -3x + 4y = 0. Solve for y in terms of x: Solve for y in terms of x: 4y = 3x, y = 3x / 4. Substitute back into total: Substitute y = 3x / 4 back into the total number of soldiers: x + 3x / 4 = 7k (where k is a scaling factor). Simplify to 7x / 4 = 7k. Find x and y values: Solve for x: x = 4k. Then, y = 3k. Calculate fraction of combat soldiers: Find the fraction of combat soldiers: x / (x + y) = 4k / (4k + 3k) = 4 / 7.

$4$ sets of uniforms were issued to each combat soldier and $3$ sets of uniforms were issued to each non-combat soldier in an army camp. The ratio of the number of soldiers in the camp to the number of uniforms issued was $7:25$ . What fraction of the soldiers in the army camp were combat soldiers?

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