Rational Numbers and Bar Graph Problem Analysis

Bar Graph Analysis: Greengrocer Produce Masses

• Graph Title: Grafik: Manavda Satılan Ürünlerin Kütleleri (Quantities of Products Sold at the Greengrocer).
• Vertical Axis: Ürün Miktarı ($\text{kg}$) — representing the mass of each sold item in kilograms.
• Horizontal Axis: Specific categories of produce sold:
  • Domates (Tomato)
  • Salatalık (Cucumber)
  • Biber (Pepper)
  • Patlıcan (Eggplant)
• Relative Heights of Bars:
  • Domates (Red Bar): Lowest bar on the graph, aligned with the lowest horizontal reference line.
  • Patlıcan (Purple Bar): Second-lowest bar, reaching the second horizontal reference line from the bottom.
  • Biber (Orange Bar): Second-tallest bar, aligned with the third horizontal reference line from the bottom.
  • Salatalık (Green Bar): Tallest bar, reaching the uppermost horizontal reference line.
• Inequality of Bar Heights:$\text{Domates} < \text{Patlıcan} < \text{Biber} < \text{Salatalık}$

Rational Number Comparison and Weight Matching

• Given Quantities (in kilograms):
  • $7\frac{2}{3}\,\text{kg}$
  • $\frac{17}{4}\,\text{kg}$
  • $6\frac{8}{9}\,\text{kg}$
  • $\frac{21}{5}\,\text{kg}$
• Conversions to Decimal and Common Forms:
  • $7\frac{2}{3} = \frac{(7 \times 3) + 2}{3} = \frac{23}{3} = 7.\overline{6}\,\text{kg} \approx 7.667\,\text{kg}$
  • $\frac{17}{4} = 4\frac{1}{4} = 4.25\,\text{kg}$
  • $6\frac{8}{9} = \frac{(6 \times 9) + 8}{9} = \frac{62}{9} = 6.\overline{8}\,\text{kg} \approx 6.889\,\text{kg}$
  • $\frac{21}{5} = 4\frac{1}{5} = 4.20\,\text{kg}$
• Pairwise and Global Ordering:
  • Comparing the values between $4$ and $5$ ($\frac{21}{5}$ and $\frac{17}{4}$):
  • Finding the common denominator ($\text{LCM}(5, 4) = 20$):       $\frac{21}{5} = \frac{21 \times 4}{5 \times 4} = \frac{84}{20}$$\frac{17}{4} = \frac{17 \times 5}{4 \times 5} = \frac{85}{20}$
  • Since $84 < 85$, it follows that:       $\frac{21}{5} < \frac{17}{4}$
  • Comparing the remaining two values:
  • Integer parts show $6 < 7$, so:       $6\frac{8}{9} < 7\frac{2}{3}$
  • Complete ascending order of all four quantities:     $\frac{21}{5} < \frac{17}{4} < 6\frac{8}{9} < 7\frac{2}{3}$$4.20\,\text{kg} < 4.25\,\text{kg} < 6.\overline{8}\,\text{kg} < 7.\overline{6}\,\text{kg}$
• Produce Quantity Assignments:
  • Domates: $\frac{21}{5}\,\text{kg}$ ($4.2\,\text{kg}$) — matches the smallest bar.
  • Patlıcan: $\frac{17}{4}\,\text{kg}$ ($4.25\,\text{kg}$) — matches the second-smallest bar.
  • Biber: $6\frac{8}{9}\,\text{kg}$ ($\approx 6.89\,\text{kg}$) — matches the second-tallest bar.
  • Salatalık: $7\frac{2}{3}\,\text{kg}$ ($\approx 7.67\,\text{kg}$) — matches the tallest bar.
• Question Objective and Final Result:
  • The question asks for the amount of tomatoes (domates) sold in kilograms.
  • Result: Domates corresponds to $\frac{21}{5}\,\text{kg}$.

Adjacent Problem: Soil Layer Distances and Elevation

• Problem Overview:
  • Involves points labeled $A$, $B$, and $C$ situated at specific underground or surface depths/elevations shown in meters across colored geological layers.
  • Requires finding the rational values corresponding to distances/depths for each point $A$, $B$, and $C$.
• Column $A$ Values in Answer Choices:
  • Option A: $\frac{1}{2}$ (equivalent to $\frac{4}{8}$)
  • Option B: $\frac{5}{8}$
  • Option C: $\frac{3}{4}$ (equivalent to $\frac{6}{8}$)
  • Option D: $\frac{1}{2}$

Adjacent Problem: Table of Rational Numbers

• Visible Column Entries (under blue/purple circular marker):
  • $\frac{11}{12}$
  • $\frac{5}{6}$
  • $\frac{2}{3}$
  • $\frac{2}{3}$