Grade 4 Math Logical Reasoning worksheet with four problems on number patterns, estimation, block counting, and missing values.

Let's solve each problem step by step.

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Problem 1: Number Machine

Question:
When a number is put into the machine, a different number comes out. If 4 goes in, 2 comes out. If 6 goes in, 3 comes out. If 8 goes in, 4 comes out. If 12 goes in, what number should come out?

#### Solution:
1. Analyze the given inputs and outputs:
- Input: 4 → Output: 2
- Input: 6 → Output: 3
- Input: 8 → Output: 4

2. Identify the pattern:
- For input 4, output is \( \frac{4}{2} = 2 \).
- For input 6, output is \( \frac{6}{2} = 3 \).
- For input 8, output is \( \frac{8}{2} = 4 \).

The pattern suggests that the output is half of the input.

3. Apply the pattern to the input 12:
- Output = \( \frac{12}{2} = 6 \).

#### Answer:
\[
\boxed{6}
\]

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Problem 2: Estimating the Product

Question:
Kevin roughly estimates the product \( 52 \times 298 \) as follows:
- He rounded each number to the nearest ten.
- Then he multiplied the numbers.
What is his estimate?

#### Solution:
1. Round each number to the nearest ten:
- \( 52 \) rounds to \( 50 \).
- \( 298 \) rounds to \( 300 \).

2. Multiply the rounded numbers:
\[
50 \times 300 = 15000
\]

#### Answer:
\[
\boxed{15000}
\]

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Problem 3: Counting Blocks

Question:
The image shows a sequence of figures with blocks. Determine how many blocks will be in the next figure.

#### Solution:
1. Analyze the pattern in the number of blocks:
- Figure 1: 1 block
- Figure 2: 3 blocks
- Figure 3: 6 blocks
- Figure 4: 10 blocks

2. Identify the pattern:
- The number of blocks in each figure forms a sequence: \( 1, 3, 6, 10 \).
- This sequence is the sequence of triangular numbers, where the \( n \)-th term is given by:
\[
T_n = \frac{n(n+1)}{2}
\]

3. Calculate the number of blocks in the next figure (Figure 5):
- For \( n = 5 \):
\[
T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = 15
\]

#### Answer:
\[
\boxed{15}
\]

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Problem 4: Finding the Missing Value

Question:
If the sum of both diagonals are the same, find the missing value.

#### Solution:
1. Represent the grid with the given values:
\[
\begin{array}{ccc}
\frac{8}{19} & & \frac{8}{19} \\
& \frac{9}{19} & \\
\frac{5}{19} & & \frac{5}{19} \\
& \frac{3}{19} & \\
\frac{6}{19} & & \frac{4}{19} \\
& x & \\
\end{array}
\]

2. Identify the diagonals:
- Main diagonal (top-left to bottom-right): \( \frac{8}{19}, \frac{9}{19}, \frac{3}{19}, x \)
- Secondary diagonal (top-right to bottom-left): \( \frac{8}{19}, \frac{5}{19}, \frac{4}{19}, \frac{6}{19} \)

3. Calculate the sum of the secondary diagonal:
\[
\frac{8}{19} + \frac{5}{19} + \frac{4}{19} + \frac{6}{19} = \frac{8 + 5 + 4 + 6}{19} = \frac{23}{19}
\]

4. Set the sum of the main diagonal equal to the sum of the secondary diagonal:
\[
\frac{8}{19} + \frac{9}{19} + \frac{3}{19} + x = \frac{23}{19}
\]

5. Simplify the left side:
\[
\frac{8 + 9 + 3}{19} + x = \frac{23}{19}
\]
\[
\frac{20}{19} + x = \frac{23}{19}
\]

6. Solve for \( x \):
\[
x = \frac{23}{19} - \frac{20}{19} = \frac{3}{19}
\]

#### Answer:
\[
\boxed{\frac{3}{19}}
\]

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Final Answers:

1. \(\boxed{6}\)
2. \(\boxed{15000}\)
3. \(\boxed{15}\)
4. \(\boxed{\frac{3}{19}}\)