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7. determine the pattern, then write the decimals to complete the decim…

Question

  1. determine the pattern, then write the decimals to complete the decimal grid.

0.34
0.65

Explanation:

Step1: Analyze vertical pattern (middle column)

Assume middle column increases by 0.1 each time (from 0.34). Let's check the vertical distance from 0.34 to 0.65. The number of steps: from row 2 (0.34) to row 5 (0.65), that's 3 steps. The difference is \(0.65 - 0.34 = 0.31\), which doesn't fit 0.1 per step. Maybe horizontal/vertical increments of 0.1 in tens and units? Wait, 0.34: 3 tenths, 4 hundredths. Let's look at horizontal (same row) or vertical (same column) changes. Let's assume vertical column (middle) has numbers: 0.34, then next (row3) \(0.34 + 0.1 = 0.44\), row4 \(0.44 + 0.1 = 0.54\), row5 (but 0.65 is in right column, row5). Wait, right column: top (row1) to row5 (0.65). Let's see the middle number is 0.34 (row2). Let's check horizontal: left, middle, right. Maybe left column: row1, row2 (empty), row3, row4 (empty), row5. Middle column: row1 (empty), row2 (0.34), row3, row4, row5 (empty). Right column: row1, row2 (empty), row3, row4 (empty), row5 (0.65). Wait, maybe the pattern is that in each column, the decimal increases by 0.1 per row down, and in each row, increases by 0.01 per column right? Wait, 0.34: 3 tenths, 4 hundredths. Let's try vertical (column) increments of 0.1 (tenths place +1) and horizontal (row) increments of 0.01 (hundredths place +1).

Step2: Fill middle column (vertical)

Middle column (column 2):

  • Row1: \(0.34 - 0.1 = 0.24\) (since row2 is 0.34, row1 is above, so -0.1)
  • Row3: \(0.34 + 0.1 = 0.44\)
  • Row4: \(0.44 + 0.1 = 0.54\)
  • Row5: \(0.54 + 0.1 = 0.64\) (but right column row5 is 0.65, so maybe hundredths place +1 for right column)

Step3: Fill right column (column 3)

Right column:

  • Row1: Let's see row1, column3. Middle column row1 is 0.24, so right column row1: \(0.24 + 0.01 = 0.25\) (hundredths +1)
  • Row2: \(0.34 + 0.01 = 0.35\)
  • Row3: \(0.44 + 0.01 = 0.45\)
  • Row4: \(0.54 + 0.01 = 0.55\)
  • Row5: 0.65 (given)

Step4: Fill left column (column 1)

Left column:

  • Row1: Middle column row1 is 0.24, so left column row1: \(0.24 - 0.01 = 0.23\) (hundredths -1)
  • Row2: \(0.34 - 0.01 = 0.33\)
  • Row3: \(0.44 - 0.01 = 0.43\)
  • Row4: \(0.54 - 0.01 = 0.53\)
  • Row5: \(0.64 - 0.01 = 0.63\) (since middle column row5 is 0.64, left column row5 is 0.63)

Wait, let's verify the right column row5: 0.65. Middle column row5: 0.64, right column row5: 0.64 + 0.01 = 0.65 (correct). Middle column row2: 0.34, row5: 0.64 (0.34 + 0.3 = 0.64? No, 0.34 to 0.64 is 0.3, 3 steps (row2 to row5: 3 rows down), so 0.1 per row. Yes! Middle column: row2 (0.34), row3 (0.44), row4 (0.54), row5 (0.64). Then right column: row5 (0.65) = 0.64 + 0.01. So right column: row1: 0.25 (0.24 + 0.01), row2: 0.35 (0.34 + 0.01), row3: 0.45 (0.44 + 0.01), row4: 0.55 (0.54 + 0.01), row5: 0.65 (correct). Left column: row1: 0.23 (0.24 - 0.01), row2: 0.33 (0.34 - 0.01), row3: 0.43 (0.44 - 0.01), row4: 0.53 (0.54 - 0.01), row5: 0.63 (0.64 - 0.01).

Step5: Fill the grid

  • Row1 (top row):
  • Left: \(0.23\)
  • Middle: \(0.24\)
  • Right: \(0.25\)
  • Row2:
  • Left: \(0.33\)
  • Middle: \(0.34\) (given)
  • Right: \(0.35\)
  • Row3:
  • Left: \(0.43\)
  • Middle: \(0.44\)
  • Right: \(0.45\)
  • Row4:
  • Left: \(0.53\)
  • Middle: \(0.54\)
  • Right: \(0.55\)
  • Row5 (bottom row):
  • Left: \(0.63\)
  • Middle: \(0.64\)
  • Right: \(0.65\) (given)

Let's check the pattern: In each row, left < middle < right (difference 0.01). In each column, top < bottom (difference 0.1). For example, left column: 0.23 (row1), 0.33 (row2, +0.1), 0.43 (row3, +0.1), 0.53 (row4, +0.1), 0.63 (row5, +0.1). Middle column: 0.24, 0.34, 0.44, 0.54, 0.64 (each +0.1). Right co…

Answer:

The filled grid (rows from top to bottom, columns left to right):

Row 1: \(0.23\), \(0.24\), \(0.25\)
Row 2: \(0.33\), \(0.34\), \(0.35\)
Row 3: \(0.43\), \(0.44\), \(0.45\)
Row 4: \(0.53\), \(0.54\), \(0.55\)
Row 5: \(0.63\), \(0.64\), \(0.65\)

(Note: The empty boxes in the original grid are filled with these decimals following the pattern of tenths increasing by 0.1 per row down and hundredths increasing by 0.01 per column right.)