A company received two shipments of ball bearings. In the first shipment, $1$ percent of the ball bearings were defective. In the second shipment, which was twice as large as the first, $4.5$ percent of the ball bearings were defective. If the company received a total of $100$ defective ball bearings, how many ball bearings were in the first shipment?

### Concept & Logic This problem requires setting up a basic linear equation based on parts of a whole. Establish a base variable for the unknown size of the first shipment, express the second shipment in terms of that base, and sum their respective defective amounts to equal the total given. ### Step-by-step Solution **Step 1: Define the variables** Let the total number of ball bearings in the first shipment be $x$. Since the second shipment is twice as large, its total number is $2x$. **Step 2: Express the defective amounts for each shipment** Defectives in 1st shipment $= 1\%$ of $x = 0.01x$ Defectives in 2nd shipment $= 4.5\%$ of $2x = 0.045 \times 2x = 0.09x$ **Step 3: Set up the total defective equation** We are given that the total number of defective ball bearings is $100$. $$0.01x + 0.09x = 100$$ **Step 4: Solve for x** Combine the like terms: $0.10x = 100$ Isolate $x$ by dividing by $0.10$ (which is the same as multiplying by $10$): $x = \frac{100}{0.10}$ $x = 1000$ The first shipment ($x$) contained $1000$ ball bearings. ### Exam Strategy & Shortcut Use a weighted average approach mentally. Assume the first shipment is $100$ units. It yields $1$ defect. The second shipment is $200$ units ($2 \times$ larger). It yields $4.5\%$ of $200 = 9$ defects. Total defects for every $300$ shipped $= 1 + 9 = 10$ defects. Since the actual total defects are $100$ (which is $10 \times 10$), the actual shipment sizes must be $10$ times our assumed numbers. First shipment $= 100 \times 10 = 1000$. ### Common Pitfall Students sometimes calculate the total combined percentage incorrectly by averaging $1\%$ and $4.5\%$ to get $2.75\%$, completely ignoring that the second shipment is twice as large (it has a higher weight). Always account for the different base sizes when adding percentages. ### Final Answer **Therefore, the correct answer is 1000.**