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

$4$ 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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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?

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: $4y = 3x$ , $y = \frac{3x}{4}$ .
Substitute back into total: Substitute $y = \frac{3x}{4}$ back into the total number of soldiers: $x + \frac{3x}{4} = 7k$ (where $k$ is a scaling factor). Simplify to $\frac{7x}{4} = 7k$ .
Find $x$ and $y$ values: Solve for $x$ : $x = 4k$ . Then, $y = 3k$ .
Calculate fraction of combat soldiers: Find the fraction of combat soldiers: $x / (x + y) = 4k / (4k + 3k) = 4 / 7$ .