Find Hours at Each Job Using a System of Equations
Giselle works as a carpenter and blacksmith. She earns $20 per hour as a carpenter and $25 per hour as a blacksmith. Last week she worked both jobs for a total of 30 hours and earned $690. How many hours did she work at each job?
Already Got the Answer?
- Setup:
c + b = 30(hours) and20c + 25b = 690(dollars) - Answer: 12 hours as a carpenter, 18 hours as a blacksmith
- One-line check:
12 + 18 = 30and240 + 450 = 690✓
If your numbers match, skip ahead to the second and third methods, the pitfalls, and the four extension problems. That is where the real learning is.
What You Will Learn
- Translating one story into two equations. Every two-unknown word problem hides two separate facts. Here they are a count (hours) and a value (dollars).
- Why "rate × quantity" is the engine of value equations. The same structure drives coin, ticket, mixture, and investment problems.
- Reading the answer as a weighted average. The $23 average hourly pay tells you almost everything about the split before you do any algebra.
- Choosing between substitution, a balance (seesaw) argument, and a baseline shift. All three give the same answer, and knowing all three lets you check yourself.
- Sanity-checking with bounds. An average can never fall outside the range of the values being averaged.
Solution: Method 1 — The Substitution Approach
This problem hands you two totals: 30 hours and $690. Two totals about two unknowns almost always means a system of two equations. Substitution is the cleanest route because one equation is already nearly solved for a variable.
Step 1 — Name the unknowns
Choose letters that remind you what they stand for, and say the units out loud. Let c = hours worked as a carpenter and b = hours worked as a blacksmith. Both are measured in hours, and both must be non-negative.
Step 2 — Write the hours equation
The hours at the two jobs must add up to the total for the week.
Step 3 — Write the earnings equation
Pay from one job is rate × hours. Carpentry brings in 20c dollars and blacksmithing brings in 25b dollars. Together they must equal $690.
Notice that the first equation adds hours and the second adds dollars. They are two different statements about the same two numbers, which is why you need both.
Step 4 — Substitute
From the hours equation, c = 30 − b. Replace c in the earnings equation. The parentheses matter: the 20 multiplies the entire expression for carpenter hours.
Step 5 — Solve for b
Step 6 — Back-substitute to find c
Giselle worked 12 hours as a carpenter and 18 hours as a blacksmith.
Solution: Method 2 — The Weighted-Average (Seesaw) Approach
Before any algebra, ask: what was Giselle's average pay per hour last week? Total pay divided by total hours is $690 ÷ 30 = $23 per hour. That single number contains the whole answer.
carpenter
average
blacksmith
The $23 average sits 3 dollars above the carpenter rate and 2 dollars below the blacksmith rate.
Step 1 — Measure how far each rate is from the average
Every carpenter hour pays $3 less than the average, so it creates a $3 "shortfall." Every blacksmith hour pays $2 more than the average, so it creates a $2 "surplus."
Step 2 — Balance the seesaw
For the average to come out at exactly $23, total shortfall must equal total surplus. With c carpenter hours and b blacksmith hours:
The hours are in the opposite ratio of the distances. The blacksmith rate is closer to the average, so it gets more hours.
Step 3 — Split the 30 hours in the ratio 2 : 3
Two parts plus three parts is five parts, and 30 ÷ 5 = 6 hours per part.
Same answer as Method 1, but with no equations written at all. The seesaw argument is the "mixture" way of seeing this problem, and it works for any two rates.
Solution: Method 3 — The Baseline-Shift Approach
Start with an imagined version of last week that is easy to compute, then correct it. Suppose all 30 hours had been spent carpentering at the lower rate: 30 × $20 = $600. Giselle actually earned $690, which is $90 more.
Where does that extra $90 come from? Each hour she switches from carpentry to blacksmithing raises her pay by $25 − $20 = $5. The number of switched hours is therefore $90 ÷ $5 = 18. Those are her blacksmith hours, and the remaining 30 − 18 = 12 are carpentry hours.
You can watch this happen in a table. Each row moves one block of hours from carpenter to blacksmith:
| Blacksmith hrs | Carpenter hrs | Total pay | Compared with $690 |
|---|---|---|---|
| 0 | 30 | $600 | $90 too low |
| 10 | 20 | $650 | $40 too low |
| 18 | 12 | $690 | Exact match ✓ |
| 20 | 10 | $700 | $10 too high |
| 30 | 0 | $750 | $60 too high |
Every extra blacksmith hour adds exactly $5. That constant increase is a linear relationship, and it is the reason a unique answer exists.
The Answer
Carpenter:12 hours (earning $240)
Blacksmith:18 hours (earning $450)
Together: 30 hours and $690.
Verification
Check the answer against both original conditions, not just the one you used last.
Hours:12 + 18 = 30 ✓
Earnings:20(12) + 25(18) = 240 + 450 = 690 ✓
Independent check (average rate):690 ÷ 30 = 23, and the weighted average computed from the answer is (12·20 + 18·25) ÷ 30 = 690 ÷ 30 = 23 ✓
Sanity Check
Your answer should pass a quick plausibility test before you ever substitute. Giselle's average pay of $23 must lie between $20 and $25, since an average cannot exceed the biggest value or fall below the smallest. It does.
It is also closer to $25 than to $20, which tells you she spent more time as a blacksmith. Our answer agrees: 18 hours versus 12.
| Scenario | Hours (carp / smith) | Average pay |
|---|---|---|
| All carpentry | 30 / 0 | $20.00 |
| Even split | 15 / 15 | $22.50 |
| Giselle's week | 12 / 18 | $23.00 |
| All blacksmithing | 0 / 30 | $25.00 |
The even split gives $22.50, so a $23 average needs to tilt a little toward the blacksmith side. That tilt is exactly what 12 / 18 provides.
What Trips People Up
✗ Splitting the hours evenly.30 ÷ 2 = 15 hours each, so 20(15) + 25(15) = 300 + 375 = 675.
That is $15 short of $690. Splitting evenly assumes the two jobs contribute equally, but the earnings figure is the clue that they don't. Whenever a problem gives you both a count and a value, an even split is almost never right, and the gap between your result and the target tells you which way to adjust.
✗ Distributing incorrectly in the substitution.20(30 − b) = 600 − b, which then leads to 600 + 24b = 690.
The 20 must multiply both terms inside the parentheses: 20(30 − b) = 600 − 20b. Skipping this is the most common source of wrong answers here, and the result (b = 3.75) is a warning sign, since fractional answers are suspicious when the data are all whole numbers.
✗ Swapping which rate goes with which job. Writing 25c + 20b = 690, or assigning 18 hours to carpentry at the end.
If you swap them, you get 20(18) + 25(12) = 360 + 300 = 660, which is $30 short. Always attach each rate to the variable you defined it with, and write the pairing down: carpenter ↔ c ↔ $20.
✗ Substituting back into the same equation you solved. Solving c = 30 − b and then plugging it into c + b = 30.
This gives 30 − b + b = 30, or 30 = 30, which is true for every value of b and tells you nothing. A substitution must go into the other equation. If you ever get a statement with no variable left, that is the signal you used the same equation twice.
The Tell-Tale Signs
This is a two-variable value problem, and it has many disguises. Look for this combination:
- Two categories, each with its own rate or price (dollars per hour, cents per coin, dollars per ticket, percent concentration).
- A total count across both categories (30 hours, 50 coins, 200 tickets, 10 liters).
- A total value across both categories ($690, $4.35, $1,800, a final concentration).
- A question asking "how many of each?"
The same skeleton appears in other stories:
Same structure, different costume: "A theater sold 200 tickets, adult tickets at $12 and child tickets at $8, for a total of $2,000. How many of each?" Replace hours with tickets and wages with prices and you have Giselle's problem exactly: a + k = 200 and 12a + 8k = 2000.
The Pattern Behind This
Let the two rates be r₁ and r₂ (with r₁ < r₂), let the total hours be T, and let the total earnings be E. The system is:
Substituting x₁ = T − x₂ and simplifying gives a general formula, which is exactly the "baseline shift" of Method 3 written in symbols:
For Giselle: x₂ = (690 − 20·30) / (25 − 20) = 90 / 5 = 18. The numerator is "how much more than the all-cheap-rate baseline did she earn," and the denominator is "how much does one switched hour add."
Limits of the shortcut. It requires r₁ ≠ r₂ (otherwise you divide by zero, and the problem is either impossible or underdetermined). It also requires that E lie between r₁T and r₂T. If it doesn't, the answer would need negative hours, which signals an impossible problem. The formula also covers only two categories; with three jobs you need a third piece of information.
The deeper idea is that the average rate E/T is a weighted average of the two rates, with hours as the weights. That is the same mathematics behind mixing solutions of different concentrations, calculating a course grade from weighted categories, and finding the blended interest rate on two loans.
Beyond the Textbook
- Payroll and freelancing: A contractor billing at two different hourly rates can recover the hour split from a single invoice total and the total hours logged.
- Blended loan or investment rates: If $10,000 is split between a 3% and a 6% account and the year's interest was $510, the same system tells you how much went into each.
- Pharmacy and chemistry: Mixing a 20% and a 25% solution to get a target concentration follows the identical weighted-average logic, with volumes in place of hours.
Extend Your Thinking
Each problem below changes something meaningful. Try it on paper first, then open the solution to compare your reasoning, not just your answer.
In a busier week, Giselle earns $18 per hour as a carpenter and $26 per hour as a blacksmith. She works 40 hours in total and earns $840. How many hours does she work at each job?
Let c = carpenter hours and b = blacksmith hours.
Hours: c + b = 40. Earnings: 18c + 26b = 840.
Use c = 40 − b: 18(40 − b) + 26b = 840, so 720 − 18b + 26b = 840, which gives 720 + 8b = 840.
8b = 120, so b = 15. Then c = 40 − 15 = 25. She worked 25 hours as a carpenter and 15 hours as a blacksmith.
Hours: 25 + 15 = 40 ✓. Pay: 18(25) + 26(15) = 450 + 390 = 840 ✓.
With rates of $20 per hour (carpenter) and $25 per hour (blacksmith), Giselle worked 40 hours one week and her average pay came out to exactly $23.50 per hour. How many hours did she work at each job?
Average × hours gives total earnings: 23.50 × 40 = 940 dollars.
With c carpenter hours and b blacksmith hours: c + b = 40 and 20c + 25b = 940.
20(40 − b) + 25b = 940 becomes 800 + 5b = 940, so 5b = 140 and b = 28. Then c = 40 − 28 = 12.
$23.50 is $3.50 above $20 and $1.50 below $25. The hour ratio is 1.5 : 3.5 = 3 : 7 (carpenter : blacksmith), and 40 ÷ 10 = 4 per part gives 12 and 28.
20(12) + 25(28) = 240 + 700 = 940 and 940 ÷ 40 = 23.50 ✓. 12 hours as a carpenter and 28 hours as a blacksmith.
Giselle picks up a third job as a glassblower at $35 per hour, still earning $20 per hour as a carpenter and $25 per hour as a blacksmith. One week she works 40 hours across the three jobs, spends the same number of hours carpentering as glassblowing, and earns $1,050. How many hours does she work at each job?
Let c = carpenter hours, b = blacksmith hours, g = glassblower hours. The clue "same hours carpentering and glassblowing" means g = c, which reduces three unknowns to two.
c + b + c = 40, so 2c + b = 40 and b = 40 − 2c.
20c + 25b + 35c = 1050, so 55c + 25b = 1050.
55c + 25(40 − 2c) = 1050 becomes 55c + 1000 − 50c = 1050, so 5c + 1000 = 1050 and c = 10. Then g = 10 and b = 40 − 20 = 20.
Hours: 10 + 20 + 10 = 40 ✓. Pay: 20(10) + 25(20) + 35(10) = 200 + 500 + 350 = 1050 ✓. 10 hours carpentering, 20 hours blacksmithing, 10 hours glassblowing.
Back to the original rates, with one twist: Giselle's blacksmith hours beyond the first 15 in a week are paid at time-and-a-half ($37.50 per hour). One week she works 30 hours in total across both jobs and earns $762.50. How many hours does she work at each job?
If she worked 15 or fewer blacksmith hours, the maximum pay would be at b = 15: 20(15) + 25(15) = 675, which is below $762.50. So she must have worked more than 15 blacksmith hours, and the overtime rate applies.
For b > 15, blacksmith pay is 25(15) + 37.50(b − 15). Carpenter hours are c = 30 − b.
20(30 − b) + 25(15) + 37.50(b − 15) = 762.50. Expanding: 600 − 20b + 375 + 37.5b − 562.5 = 762.5.
Constants: 600 + 375 − 562.5 = 412.5. Variable terms: −20b + 37.5b = 17.5b. So 412.5 + 17.5b = 762.5, giving 17.5b = 350 and b = 20. Then c = 30 − 20 = 10. The result satisfies b > 15, so the case assumption holds.
Carpenter: 20(10) = 200. Blacksmith: 25(15) + 37.50(5) = 375 + 187.50 = 562.50. Total: 200 + 562.50 = 762.50 ✓, and 10 + 20 = 30 hours ✓. 10 hours as a carpenter and 20 hours as a blacksmith.
Frequently Asked Questions
c + b = 30 and 20c + 25b = 690. Substituting c = 30 − b gives 600 + 5b = 690, so b = 18 blacksmith hours and c = 12 carpenter hours.12 + 18 = 30 hours, $240 + $450 = $690, and the average of $23 per hour lies between $20 and $25, and is a little closer to $25 because more hours were worked at that rate.2026-06-08