6.3 Ratios, Proportions, Partitive Proportions, and Inverse/Direct Variation

6.3 Ratios, Proportions, Partitive Proportions, and Inverse/Direct Variation

Understanding the mathematical behavior of ratios and variations allows foreign service officers to model organizational dynamics. When consular volume doubles, does staffing need to double (direct variation), or does processing time per applicant decrease through efficiency gains? Mastering these principles turns complex multi-variable word problems into clean, solvable equations.

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Foundational Principles of Ratios and Proportions

A ratio compares two quantities, written $a:b$ or $a/b$ with a nonzero denominator. A dimensionless comparison of lengths requires consistent units; a ratio of unlike units is a rate, such as kilometers per hour.

A proportion is an equation stating that two ratios are equal:

$$\frac{a}{b} = \frac{c}{d} \quad \text{or} \quad a : b = c : d$$

For $a/b=c/d$ with $b,d\ne0$, cross-multiplication gives $ad=bc$. If the unknown is d, then $d=bc/a$ provided $a\ne0$. Identify the actual unknown before solving; that rearrangement does not apply to every placement of x.

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Harmonizing and Combining Linked Pairwise Ratios

A common civil service scenario provides two separate pairwise ratios sharing a common intermediate variable, requiring the candidate to find the unified three-variable ratio:

• Given: $A : B = 2 : 3$ and $B : C = 4 : 5$
• Target: Determine the compound ratio $A : B : C$.

The LCM Harmonization Method

To combine these ratios, make the shared variable ($B$) identical across both expressions by finding the Least Common Multiple (LCM) of its values:

1. In the first ratio, $B$ corresponds to $3$.
2. In the second ratio, $B$ corresponds to $4$.
3. The LCM of $3$ and $4$ is $12$.
4. Scale the first ratio by multiplying both terms by $4$: $A : B = (2 \times 4) : (3 \times 4) = 8 : 12$.
5. Scale the second ratio by multiplying both terms by $3$: $B : C = (4 \times 3) : (5 \times 3) = 12 : 15$.
6. Combine the harmonized terms: $A : B : C = 8 : 12 : 15$.

This technique extends directly to four-variable chain ratios ($A:B, B:C, C:D$) by successively harmonizing adjacent linking variables.

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Partitive Proportion in Diplomatic Resource Allocations

Partitive proportion involves partitioning a whole quantity into two or more unequal shares according to a specified ratio $a : b : c$.

General Algebraic Model

Let the common scalar multiplier (the value of one ratio part) be represented by $x$. The sum of the individual parts must equal the total quantity $T$:

$$ax + bx + cx = T$$ $$x(a + b + c) = T \implies x = \frac{T}{a + b + c}$$

Once the unit value $x$ is determined, multiply $x$ by each ratio coefficient to compute each share:

• Share $A = a \times x$
• Share $B = b \times x$
• Share $C = c \times x$

Administrative Example

A foreign ministry divides an emergency disaster response grant of $\text{PHP } 3,600,000$ among three consulates in the ratio $3 : 4 : 5$.

1. Sum the ratio parts: $3 + 4 + 5 = 12$ total parts.
2. Calculate the value of one part: $x = \frac{\text{PHP } 3,600,000}{12} = \text{PHP } 300,000$.
3. Distribute the allocations:
   • Consulate 1: $3 \times \text{PHP } 300,000 = \text{PHP } 900,000$
   • Consulate 2: $4 \times \text{PHP } 300,000 = \text{PHP } 1,200,000$
   • Consulate 3: $5 \times \text{PHP } 300,000 = \text{PHP } 1,500,000$

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Direct Variation (Direct Proportion)

Direct variation has $y=kx$. With positive k, y rises as x rises; with negative k, the direction reverses. The proportional model, rather than an observation that both quantities happen to rise, establishes direct variation.

Operational Formulation

When comparing two distinct states $(x_1, y_1)$ and $(x_2, y_2)$:

$$\frac{y_1}{x_1} = \frac{y_2}{x_2} \implies y_2 = y_1 \left(\frac{x_2}{x_1}\right)$$

• Graph: A straight line passing through the origin $(0, 0)$ with slope $k$.
• Applications: Passport revenue generated versus number of applicants; total diplomatic courier baggage fees versus total parcel weight.

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Inverse Variation (Indirect Proportion)

Inverse variation has $xy=k$, or $y=k/x$ for nonzero x. For positive x and k, increasing x reduces y. The product remains constant; an arbitrary downward trend alone does not prove inverse variation.

Operational Formulation

When comparing two distinct states:

$$x_1 y_1 = x_2 y_2 \implies \frac{x_1}{x_2} = \frac{y_2}{y_1}$$

• For positive x and k, the graph lies in the first quadrant and approaches the axes. Other sign domains give other branches.
• A fixed-work staffing model is inverse only if per-worker productivity is constant and independent, total work is fixed, and no coordination bottleneck changes the rate.

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Joint and Combined Variation in Operational Logistics

Real-world administrative challenges rarely depend on a single variable. They involve multiple factors acting simultaneously:

1. Joint Variation

A variable varies jointly as two or more variables if it is directly proportional to their product:

$$z = kxy$$

Example: Total bilateral development expenditure ($z$) varies jointly with the number of field missions ($x$) and the average days per mission ($y$).

2. Combined Variation (The Master Logistics Formula)

Combined variation occurs when a variable varies directly with one or more variables and inversely with another:

$$z = \frac{k x}{y}$$

In civil service examinations, this relationship is tested through workforce logistics problems connecting workers ($N$), days ($D$), hours per day ($H$), and output or work produced ($W$):

$$\frac{W_1}{N_1 \times D_1 \times H_1} = \frac{W_2}{N_2 \times D_2 \times H_2}$$

This unified formula eliminates guesswork: output ($W$) belongs in the numerator, while all input effort parameters ($N, D, H$) multiply in the denominator.