Question

complete the ratio table.
14 5
35
20 50
22 55
24 60

Explanation:

Step1: Find the ratio

First, check the ratio between the second column and the first column for the known rows. For example, take the row with 20 and 50: the ratio of 50 to 20 is $\frac{50}{20} = 2.5$. Let's verify with another row, 55 and 22: $\frac{55}{22} = 2.5$, and 60 and 24: $\frac{60}{24} = 2.5$. So the ratio of the second number to the first number in each row is 2.5 (or $\frac{5}{2}$). Now, for the row with 35 in the second column, let the first number be $x$. Then $\frac{35}{x} = 2.5$.

Step2: Solve for x

To find $x$, we can rearrange the equation: $x = \frac{35}{2.5}$. Calculating that, $35 \div 2.5 = 14 \times 2.5 = 35$? Wait, no, wait. Wait, actually, looking at the first row: 14 and 5. Wait, maybe I got the ratio reversed. Let's check the first row: 14 and 5. Then 5 to 14? No, wait the third row: 20 and 50. 20 is first, 50 is second. So 50/20 = 2.5. First row: 5/14? No, that doesn't match. Wait, maybe the first column is the first number, second column is the second. Wait, maybe the ratio is first number to second number? Wait 20/50 = 0.4, 22/55 = 0.4, 24/60 = 0.4. Ah! That's consistent. 20 divided by 50 is 0.4, 22 divided by 55 is 0.4, 24 divided by 60 is 0.4. So the ratio of first number to second number is 0.4 (which is 2/5). Let's check the first row: 14 and 5. 14/5 = 2.8, which is not 0.4. Wait, maybe the first row is 14 (first column) and 5 (second column), but the third row is 20 (first) and 50 (second). So 20/50 = 0.4, 22/55 = 0.4, 24/60 = 0.4. So the ratio of first to second is 0.4. Now, the row with 35 in the second column: let the first number be $x$. So $x/35 = 0.4$. Then $x = 35 \times 0.4 = 14$. Wait, no, 35 times 0.4 is 14? Wait 35 * 0.4 = 14. But the first row is 14 and 5? Wait that can't be. Wait, maybe I mixed up the columns. Let's list the rows:

Row 1: Col1:14, Col2:5

Row 2: Col1:?, Col2:35

Row 3: Col1:20, Col2:50

Row 4: Col1:22, Col2:55

Row 5: Col1:24, Col2:60

Now, let's find the ratio between Col2 and Col1 for rows 3,4,5:

Row3: 50/20 = 2.5

Row4: 55/22 = 2.5

Row5: 60/24 = 2.5

Ah! So Col2 / Col1 = 2.5. So for row3: 50/20=2.5, row4:55/22=2.5, row5:60/24=2.5. Now row1: Col2 is 5, Col1 is 14. 5/14 ≈ 0.357, which is not 2.5. Wait, maybe row1 is a typo? No, the problem is to complete the table. So row2: Col2 is 35, Col1 is x. So 35/x = 2.5. So x = 35 / 2.5 = 14? Wait no, 35 divided by 2.5 is 14? Wait 2.5 times 14 is 35. Yes! Because 2.5 * 14 = 35. Wait, row1: Col1 is 14, Col2 is 5? No, that doesn't match. Wait, maybe the first row is 14 (Col1) and 5 (Col2), but the ratio is Col2 = Col1 * k? Wait 5 = 14 * k? No, that would be k=5/14. But row3: 50 = 20 * k → k=2.5. Ah! Now I see. The first row must be a mistake? No, the table is given as 14,5; then a blank,35; then 20,50; 22,55; 24,60. So looking at the last three rows: 20 and 50 (50=20*2.5), 22 and 55 (55=22*2.5), 24 and 60 (60=24*2.5). So the ratio of Col2 to Col1 is 2.5. Now, the first row: 5=14*k? No, 14*2.5=35, which is the Col2 of the second row. Wait, maybe the first row is Col1=14, Col2=5, but that's inconsistent. Wait, no, maybe the first row is Col1=14, Col2=5, but the second row is Col1=?, Col2=35, and the ratio is Col2/Col1=5/14? No, that would be 35/(5/14)=35*(14/5)=98, which is not. Wait, no, let's check the second row: Col2=35, and we need to find Col1. Looking at the last three rows, Col1*2.5=Col2. So 20*2.5=50, 22*2.5=55, 24*2.5=60. So Col2=Col1*2.5. Now, the first row: Col2=5, so Col1=5/2.5=2? But that's not 14. Wait, maybe the first row is Col1=14, Col2=5, but that's a different ratio. Wait, maybe the table is transposed? No, the problem says "ratio table", so probably the ratio is consistent. Wait, the third row: 20 and 50. 20:50 simplifies to 2:5. 22:55 is 2:5, 24:60 is 2:5. Ah! 20:50 = 2:5, 22:55=2:5, 24:60=2:5. So the ratio of Col1 to Col2 is 2:5. So 20/50=2/5, 22/55=2/5, 24/60=2/5. Now, the first row: 14:5. 14/5=14/5, which is not 2/5. Wait, maybe the first row is a typo, or maybe I misread the table. Wait, the table is:

First column: 14, [blank], 20, 22, 24

Second column: 5, 35, 50, 55, 60

Ah! Now I see. So the rows are:

Row 1: 14 (Col1), 5 (Col2)

Row 2:? (Col1), 35 (Col2)

Row 3: 20 (Col1), 50 (Col2)

Row 4: 22 (Col1), 55 (Col2)

Row 5: 24 (Col1), 60 (Col2)

Now, let's check the ratio of Col2 to Col1 for rows 3,4,5:

50/20=2.5, 55/22=2.5, 60/24=2.5. So that's consistent (2.5). Now, row 1: 5/14≈0.357, which is not 2.5. But row 2: Col2=35, so Col1=35/2.5=14. Wait, that's the Col1 of row 1. So maybe row 1 and row 2 are related. Wait, row 1: Col1=14, Col2=5; row 2: Col1=14, Col2=35? No, that can't be. Wait, no, the blank is in Col1 of row 2, with Col2=35. So using the ratio from rows 3-5 (Col2=Col1*2.5), then Col1=Col2/2.5. So for row 2, Col2=35, so Col1=35/2.5=14. Wait, that's the same as row 1's Col1. So maybe row 1 is Col1=14, Col2=5, but that's a different ratio. But the problem is to complete the ratio table, so we have to go with the consistent ratio from the last three rows. So the ratio of Col2 to Col1 is 2.5, so Col1=Col2/2.5. For row 2, Col2=35, so Col1=35/2.5=14. Wait, but row 1 has Col1=14, Col2=5. That seems inconsistent, but maybe it's a mistake, or maybe the first row is a different ratio, but the rest follow Col2=Col1*2.5. So the blank in row 2, Col1, when Col2=35, is 35/2.5=14. Wait, but that's the same as row 1's Col1. Maybe that's correct. Let's check:

Row 1: 14, 5 (5=14*(5/14))

Row 2: 14, 35 (35=14*2.5)

Row 3: 20, 50 (50=20*2.5)

Row 4: 22, 55 (55=22*2.5)

Row 5: 24, 60 (60=24*2.5)

Ah! Now I see. The first row is a different ratio, but the rest follow Col2=Col1*2.5. So the blank in row 2, Col1, is 14, because 14*2.5=35.

Answer:

14