✅ To find out how much more Technician B earned than Technician C, we first calculate the work units completed by each technician. Work units are calculated as efficiency multiplied by the number of days worked (Work Units = Efficiency × Days).

A's work units = 2 (efficiency) × 15 (days) = 30 units.

B's work units = 3 (efficiency) × 10 (days) = 30 units.

C's work units = 4 (efficiency) × 5 (days) = 20 units.

The ratio of their earnings will be proportional to their work units: A:B:C = 30:30:20, which simplifies to 3:3:2. The total parts in this ratio are 3 + 3 + 2 = 8.

Now, we can calculate each technician's share of the total payment of $16,000. B's share = (3/8) × $16,000 = $6,000. C's share = (2/8) × $16,000 = $4,000.

The difference in earnings between Technician B and Technician C is $6,000 - $4,000 = $2,000.

❌ Option A ($1,000) is incorrect because it is not the calculated difference based on the work units and total payment. This might be a result of miscalculating individual shares.

❌ Option C ($3,000) is incorrect as it does not correspond to the actual difference in earnings. This could arise from incorrect ratio calculation or miscalculation of a technician's share.

❌ Option D ($4,000) is incorrect because this is Technician C's total earning, not the difference between B's and C's earnings.

✅ To solve this sharing and comparison problem, we first establish the proportional shares of Alex, Ben, and Chloe. Let Alex's share be represented by 3 units to avoid fractions later (since Chloe's share is 2/3 of Ben's, and Ben's is twice Alex's, a multiple of 3 for Alex simplifies calculations). Using the relationships given: Ben's share = 2 × Alex's share = 2 × 3 units = 6 units. Chloe's share = (2/3) × Ben's share = (2/3) × 6 units = 4 units.

The total ratio of shares for Alex:Ben:Chloe is 3:6:4. The difference between Chloe's share and Alex's share is 4 units - 3 units = 1 unit.

Given that this difference is $200, we know that 1 unit = $200.

The total sum of money shared is the sum of all units: Total units = 3 (Alex) + 6 (Ben) + 4 (Chloe) = 13 units.

Therefore, the total sum shared is 13 units × $200/unit = $2600.

❌ A) $2200 is incorrect. This would correspond to a total of 11 units ($2200 / $200), which doesn't match the derived total of 13 units.

❌ B) $2400 is incorrect. This would correspond to a total of 12 units ($2400 / $200). An error might lead to this if Chloe's share was miscalculated or the total units were summed incorrectly.

❌ D) $2800 is incorrect. This would correspond to a total of 14 units ($2800 / $200), which suggests an arithmetic error in summing the units.

✅ Let the common multiplier for the initial shares be 'k'. Alex's initial share = 2k, Ben's initial share = 3k, and Chris's initial share = 5k.

When Ben gives $200 to Alex, their shares change: Alex's new share = 2k + 200, and Ben's new share = 3k - 200.

The new ratio of Alex's share to Ben's share is given as 3:2, so we set up the proportion: .

Cross-multiplying yields: 2(2k + 200) = 3(3k - 200), which simplifies to 4k + 400 = 9k - 600.

Rearranging the terms to solve for k: 9k - 4k = 400 + 600, so 5k = 1000, which means k = 200.

Chris's original share was 5k, therefore Chris's share = 5 × $200 = $1000.

❌ Option A ($800) is incorrect. This would correspond to a k value of $160 if it were 5k, or 4k if k=$200, which is Alex's share if Ben gives $100 instead of $200.

❌ Option C ($1200) is incorrect. This would imply k=$240 if it were 5k, or 6k if k=$200, which doesn't fit the problem conditions.

❌ Option D ($1500) is incorrect. This would mean k=$300 if it were 5k, or 7.5k if k=$200, which is not derivable from the given ratios.

✅ First, combine the initial ratios: Given Liam (L) : Noah (N) = 2:3 and Noah (N) : Olivia (O) = 4:5. To find L:N:O, we make N common (LCM of 3 and 4 is 12). So, L:N = (2×4):(3×4) = 8:12, and N:O = (4×3):(5×3) = 12:15, which gives the combined initial ratio L:N:O = 8:12:15.

Let their initial shares be 8x, 12x, and 15x respectively. The total initial sum of money is 8x + 12x + 15x = 35x.

After Liam gives $120 to Olivia, their new shares are: Liam's new share = (8x - 120), Noah's new share = 12x (unchanged), and Olivia's new share = (15x + 120).

These new shares are in the ratio 34:60:81. We can set up a proportion using any two parts of the ratio, for example, Noah's and Liam's new shares: = .

Simplifying the proportion, = . Cross-multiplying gives 34x = 5(8x - 120), which expands to 34x = 40x - 600. Solving for x: 600 = 40x - 34x => 6x = 600 => x = 100.

Finally, the total sum of money initially shared was 35x = 35 * $100 = $3500.

❌ Options A, C, and D are incorrect as they result from miscalculations during the multi-step process, such as errors in combining ratios, setting up the proportional equations, or solving for x.

✅ Let X, Y, and Z represent the shares of the three colleagues.

From the problem statement: 1. X = (2/3)Y 2. Z = 1.5 × X = (3/2) × X

Substitute X from (1) into (2): Z = (3/2) × (2/3)Y = Y. So, the ratio of their shares can be expressed as: X : Y : Z (2/3)Y : Y : Y

To convert this to a whole number ratio, multiply all parts by 3: 2Y : 3Y : 3Y Thus, X:Y:Z = 2:3:3.

The total ratio parts are 2 + 3 + 3 = 8 parts.

We are given that Y receives $600 more than X. In terms of ratio parts, Y's share (3 parts) - X's share (2 parts) = 1 part. So, 1 part = $600.

The total bonus amount is the total number of parts multiplied by the value of one part: Total Bonus = 8 parts × $600/part = $4800.

❌ Option A ($3600) would be 6 parts, which might result from miscalculating the ratio of Z or the total parts.

❌ Option B ($4200) would be 7 parts, which could occur if Z's share was miscalculated to be 2 parts instead of 3 (e.g., X:Y:Z = 2:3:2).

❌ Option D ($5400) would be 9 parts, possibly from an incorrect calculation of Z's share relative to Y or an error in total parts summation.