Question

there is a number line and some multiplication operation cards (×9/5, ×5, ×5/9, ×9) and some numbers (-10, 20, 40, -10, 30, 50, -50, -18, 54) marked on the image. the problem is likely related to number operations and number line understanding, maybe to match the numbers with the results of the multiplication operations or analyze the positions on the number line.

Explanation:

Step1: Analyze the first number line

The first number line has -10, 20, 40. Let's see the relationship between the second number line's numbers (-10, 30, 50) and the operation. Let's take 20 to 30: 20 * \( \frac{3}{2} \)? Wait, no, let's check the multipliers. Wait, maybe we take a number from the first line and apply a multiplier to get the second line. Let's take 20 (first line middle) and 30 (second line middle): 20 * \( \frac{3}{2} \)? No, 20 * \( \frac{3}{2} \) is 30? Wait 20 * 1.5 = 30. But the multipliers are \( \times \frac{9}{5} \), \( \times 5 \), \( \times \frac{5}{9} \), \( \times 9 \). Wait, maybe another approach. Let's take -10: it's the same in both lines. Then 20 (first) to 30 (second): 20 * \( \frac{3}{2} \)? No, 20 * \( \frac{3}{2} \) is 30, but the multipliers don't have \( \frac{3}{2} \). Wait, 40 (first) to 50 (second): 40 * \( \frac{5}{4} \) = 50. Wait, 5/4 is not a multiplier here. Wait, maybe the numbers are being multiplied by a fraction. Wait, let's check the bottom numbers: -50, -18, 54. Wait, maybe we take a number and multiply by a fraction to get another. Wait, let's take 20 (first line) and multiply by \( \frac{9}{5} \): 20 * \( \frac{9}{5} \) = 36? No. 20 * \( \frac{3}{2} \) is 30. Wait, 20 * \( \frac{3}{2} \) = 30, which is the middle number in the second line. 40 * \( \frac{5}{4} \) = 50, which is the right number. But the multipliers given are \( \times \frac{9}{5} \), \( \times 5 \), \( \times \frac{5}{9} \), \( \times 9 \). Wait, maybe the first line numbers: -10, 20, 40. Second line: -10, 30, 50. The difference between 20 and 30 is 10, 40 and 50 is 10. Wait, no, maybe it's a multiplication by \( \frac{3}{2} \), but that's not a given multiplier. Wait, maybe I'm misunderstanding. Wait, the bottom has numbers -50, -18, 54. Let's see: -10 * 5 = -50. Oh! -10 * 5 = -50, which is one of the bottom numbers. 20 * \( \frac{9}{5} \) = 36? No. Wait, 20 * \( \frac{9}{5} \) = 36, not there. Wait, 40 * \( \frac{9}{5} \) = 72? No. Wait, -10 * \( \frac{9}{5} \) = -18. Oh! -10 * \( \frac{9}{5} \) = -18, which is a bottom number. And 20 * \( \frac{9}{5} \) = 36? No, 20 * \( \frac{9}{5} \) = 36, not 30. Wait, 6 * 9 = 54? No. Wait, 40 * \( \frac{9}{5} \) = 72? No. Wait, 6 * 9 = 54, but where is 6? Wait, maybe the numbers on the first line: -10, 20, 40. Let's multiply -10 by 5: -50 (bottom number). Multiply 20 by \( \frac{9}{5} \): 20 * \( \frac{9}{5} \) = 36? No. Wait, 40 * \( \frac{9}{5} \) = 72? No. Wait, -10 * \( \frac{9}{5} \) = -18 (bottom number). 20 * \( \frac{9}{5} \) = 36? No, 20 * \( \frac{9}{5} \) = 36, not 30. Wait, 6 * 9 = 54. Wait, maybe 40 * \( \frac{9}{5} \) = 72? No. Wait, maybe the middle number: 20 (first) to 30 (second) is 20 * 1.5 = 30, but 1.5 is 3/2. Not a given multiplier. Wait, maybe the operation is to take a number from the first line and multiply by a fraction to get a bottom number. Let's see: -10 * 5 = -50 (bottom left), -10 * \( \frac{9}{5} \) = -18 (bottom middle), and 40 * \( \frac{9}{5} \) = 72? No, 54 is there. Wait, 6 * 9 = 54. Wait, maybe 40 * \( \frac{9}{5} \) = 72, no. Wait, 20 * \( \frac{9}{5} \) = 36, no. Wait, maybe the right number: 40 (first) to 50 (second) is 40 * 1.25 = 50, which is 5/4. Not a multiplier. Wait, maybe I made a mistake. Let's check the multipliers again. The multipliers are \( \times \frac{9}{5} \), \( \times 5 \), \( \times \frac{5}{9} \), \( \times 9 \). Let's take -10: -10 * 5 = -50 (matches bottom left), -10 * \( \frac{9}{5} \) = -18 (matches bottom middle), and 20 * \( \frac{9}{5} \) = 36? No, 54 is there. Wait, 40 * \( \frac{9}{5} \) = 72? No. Wait, 6 * 9 = 54. Wait, maybe 40 * \( \frac{9}{5} \) is 72, no. Wait, 20 * \( \frac{9}{5} \) = 36, 40 * \( \frac{9}{5} \) = 72. Not 54. Wait, 54 divided by 9 is 6. No. Wait, maybe the numbers are -10, 20, 40. Let's multiply 20 by \( \frac{9}{5} \): 20 * \( \frac{9}{5} \) = 36. No. Wait, 40 * \( \frac{9}{5} \) = 72. No. Wait, 54 is 6 * 9. Wait, maybe 40 * \( \frac{9}{5} \) is 72, no. Wait, maybe the operation is to take a number from the first line and multiply by a fraction to get a number in the second line. Wait, 20 (first) * \( \frac{3}{2} \) = 30 (second), 40 * \( \frac{5}{4} \) = 50 (second). But the multipliers don't have \( \frac{3}{2} \) or \( \frac{5}{4} \). Wait, maybe the problem is to find which multiplier transforms the first line numbers to the second line numbers. Let's check each multiplier:

- \( \times \frac{9}{5} \): 20 * \( \frac{9}{5} \) = 36 (not 30), 40 * \( \frac{9}{5} \) = 72 (not 50)
- \( \times 5 \): 20 * 5 = 100 (not 30), 40 * 5 = 200 (not 50)
- \( \times \frac{5}{9} \): 20 * \( \frac{5}{9} \) ≈ 11.11 (not 30), 40 * \( \frac{5}{9} \) ≈ 22.22 (not 50)
- \( \times 9 \): 20 * 9 = 180 (not 30), 40 * 9 = 360 (not 50)

Wait, this is confusing. Maybe the numbers on the first line are -10, 20, 40, and the second line are -10, 30, 50. The difference between 20 and 30 is 10, 40 and 50 is 10. So adding 10? But that's not a multiplication. Wait, maybe the bottom numbers are results of multiplying -10, 20, 40 by some of the multipliers. Let's check:

- -10 * 5 = -50 (bottom left)
- -10 * \( \frac{9}{5} \) = -18 (bottom middle)
- 40 * \( \frac{9}{5} \) = 72 (not 54)
- 20 * \( \frac{9}{5} \) = 36 (not 54)
- 40 * \( \frac{9}{5} \) = 72, 20 * 9 = 180, 40 * 9 = 360. Wait, 54: 6 * 9 = 54. Where is 6? Maybe 20 * \( \frac{9}{5} \) = 36, no. Wait, maybe the first line numbers are -10, 20, 40, and we multiply 20 by \( \frac{9}{5} \) to get 36, but that's not matching. Wait, maybe the problem is to find which multiplier when applied to a number gives another. Wait, maybe the correct multiplier is \( \times \frac{9}{5} \) for -10: -10 * \( \frac{9}{5} \) = -18 (which is a bottom number), and 40 * \( \frac{9}{5} \) = 72? No, 54 is there. Wait, 54 divided by 9 is 6, 54 divided by \( \frac{9}{5} \) is 30. Oh! 30 * \( \frac{9}{5} \) = 54? Wait, 30 * \( \frac{9}{5} \) = 54. Yes! 30 * \( \frac{9}{5} \) = (30/5)*9 = 6*9 = 54. So 30 (second line middle) multiplied by \( \frac{9}{5} \) gives 54 (bottom right). And -10 * 5 = -50 (bottom left), -10 * \( \frac{9}{5} \) = -18 (bottom middle). So the multipliers are being applied to the second line numbers? Wait, the first line has 20, second line has 30. 30 * \( \frac{9}{5} \) = 54. So maybe the operation is to take a number from the second line and multiply by \( \frac{9}{5} \) to get 54. Let's check: 30 * \( \frac{9}{5} \) = 54. Yes! 30 * \( \frac{9}{5} \) = (30 ÷ 5) * 9 = 6 * 9 = 54. So the multiplier \( \times \frac{9}{5} \) when applied to 30 (from the second line) gives 54 (bottom right). Also, -10 * 5 = -50 (bottom left), -10 * \( \frac{9}{5} \) = -18 (bottom middle). So the correct multiplier for getting 54 from 30 is \( \times \frac{9}{5} \).

Step2: Confirm

Let's verify: 30 (second line middle number) multiplied by \( \frac{9}{5} \):

\( 30 \times \frac{9}{5} = \frac{30 \times 9}{5} = \frac{270}{5} = 54 \), which matches the bottom right number (54).

Answer:

The correct multiplier is \( \times \frac{9}{5} \) (the first multiplier option).