Work Rate Problem: Combined Surveyor Efficiency

What This Problem Teaches

• Combined work rate calculation: How to add individual work rates to find the efficiency of a team working together
• Rate vs. time relationship: Understanding that work rates are the reciprocals of completion times
• Fraction addition with unlike denominators: Finding common denominators to combine fractional rates
• Reciprocal reasoning: Converting from a combined rate back to the time needed for completion
• Real-world collaboration modeling: How mathematical models predict the efficiency gains of teamwork

Solution: Method 1 — The Rate Addition Approach

Step 1 — Find individual work rates

The key insight is to convert each worker's completion time into a rate (fraction of job per hour):

Step 2 — Combine the work rates

When working together, their individual contributions add up. The combined rate is:

Step 3 — Add the fractions

To add fractions with different denominators, find the least common multiple of 5 and 6, which is 30:

Step 4 — Convert rate back to time

If they complete 11/30 of the job in 1 hour, how long for the complete job? Take the reciprocal:

Step 5 — Convert to hours and minutes

Converting the improper fraction to a mixed number:

Solution: Method 2 — The Work Completion Table

Step 1 — Set up the work completion table

Instead of working with rates directly, let's track how much work each person completes in the same time period:

Step 2 — Find when total work equals 1

We need to find time t when the total work equals 1 complete job:

Step 3 — Verify using the table logic

At t = 30/11 hours, the experienced surveyor completes (1/5) × (30/11) = 6/11 of the job, and the apprentice completes (1/6) × (30/11) = 5/11 of the job. Together: 6/11 + 5/11 = 11/11 = 1 complete job. ✓

Verification

Let's check our answer by substituting back into the original problem structure:

We can also verify this makes intuitive sense: both workers together should finish faster than either alone. Since the faster worker takes 5 hours alone, and the slower takes 6 hours, finishing in about 2.7 hours seems reasonable for their combined effort.

Watch Out For These

How to Spot This Problem Type

Work rate problems typically have these telltale signs:

• "Working together" or "combined effort" — signals that you need to add individual rates
• Two or more workers with different completion times — each worker's time converts to their individual rate
• "How long will it take" when multiple workers collaborate — you're looking for the reciprocal of the combined rate
• Context involving production, construction, or task completion — manufacturing widgets, painting houses, filling pools, or surveying land
• Individual times given, combined time unknown — the classic setup for rate addition problems

This same mathematical structure appears in electrical engineering (parallel resistors), network computing (parallel processing), and pharmacology (drug clearance rates). Learning to recognize the pattern serves you across many fields.

The General Formula

For any two-worker collaboration problem, the pattern is always the same:

This simplifies to the more memorable form:

In our surveyor problem: (5 × 6) ÷ (5 + 6) = 30 ÷ 11 = 30/11 hours

Real Applications

• Construction project management: Estimating completion times when multiple crews with different productivity rates work on the same job site
• Network engineering: Calculating data transfer speeds when multiple connections work in parallel to download the same file
• Manufacturing optimization: Determining production schedules when different assembly lines with varying throughputs collaborate on the same order
• Medical pharmacology: Modeling how multiple elimination pathways (kidney and liver) work together to clear drugs from the body

Sanity Check

Our answer of 2 hours and 44 minutes passes several reasonableness tests:

Our answer falls exactly where it should: faster than either worker alone, but between the "both experienced" and "both apprentice" scenarios. The collaboration provides genuine efficiency gains.