To find the missing terms in the series \( \frac{H}{16}, \frac{K}{13}, ?, \frac{Q}{19}, \frac{T}{40}, ?, \frac{Z}{52} \), we will analyze the pattern in the series step by step.
### Step 1: Identify the Place Values
First, we note the place values of the letters:
- \( H = 8 \)
- \( K = 11 \)
- \( Q = 17 \)
- \( T = 20 \)
- \( Z = 26 \)
### Step 2: Analyze the Denominators
Next, we look at the denominators:
- The first term is \( 16 \).
- The second term is \( 13 \).
- The fourth term is \( 19 \).
- The fifth term is \( 40 \).
- The seventh term is \( 52 \).
### Step 3: Identify the Pattern in Place Values
From \( H \) to \( K \):
- Place value of \( K \) (11) is \( 3 \) more than \( H \) (8).
From \( K \) to the missing term:
- We add \( 3 \) to \( K \) (11), which gives us \( 14 \). The letter corresponding to \( 14 \) is \( N \).
From \( N \) to \( Q \):
- Place value of \( Q \) (17) is \( 3 \) more than \( N \) (14).
From \( Q \) to \( T \):
- Place value of \( T \) (20) is \( 3 \) more than \( Q \) (17).
### Step 4: Identify the Pattern in Denominators
Now, let's analyze the denominators:
- From \( 16 \) to \( 13 \): \( 16 - 3 = 13 \)
- From \( 13 \) to the missing term: We will find the next term.
- From \( 19 \) to \( 40 \): \( 19 \times 2 = 38 \), but we see \( 40 \) here, which is \( 2 \) more than \( 38 \).
- From \( 40 \) to \( 52 \): \( 40 + 12 = 52 \).
### Step 5: Calculate the Missing Denominators
Now we can fill in the missing denominators:
- The missing denominator after \( 13 \) should be \( 28 \) (as \( 13 + 15 = 28 \)).
- The second missing denominator after \( 40 \) should be \( 25 \) (as \( 40 - 15 = 25 \)).
### Step 6: Write the Missing Terms
Now we can write the missing terms:
1. The first missing term is \( \frac{N}{28} \).
2. The second missing term is \( \frac{W}{25} \).
### Final Answer
The complete series is:
- \( \frac{H}{16}, \frac{K}{13}, \frac{N}{28}, \frac{Q}{19}, \frac{T}{40}, \frac{W}{25}, \frac{Z}{52} \).
Thus, the missing terms are:
1. \( N/28 \)
2. \( W/25 \)
