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Sagot :
To solve the problem of connecting equivalent measurements, let's convert each given measurement to a consistent unit and then identify which of the provided equivalences match.
### Conversion Steps and Equivalences
#### 1. Convert \( 16 \frac{1}{2} \) cups to fluid ounces.
First, convert the mixed number to an improper fraction or decimal:
[tex]\[ 16 \frac{1}{2} = 16 + \frac{1}{2} = 16.5 \text{ cups} \][/tex]
Next, convert cups to fluid ounces:
[tex]\[ 16.5 \text{ cups} \times 8 \text{ fl ounces/cup} = 132 \text{ fl ounces} \][/tex]
#### 2. Convert 196 cups to gallons.
Convert cups to gallons knowing that 1 gallon = 16 cups:
[tex]\[ 196 \text{ cups} \div 16 \text{ cups/gallon} = 12.25 \text{ gallons} \][/tex]
#### 3. Convert \( 12 \frac{1}{2} \) pints to gallons.
Convert the mixed number to an improper fraction or decimal:
[tex]\[ 12 \frac{1}{2} = 12.5 \text{ pints} \][/tex]
Next, convert pints to gallons, knowing that 1 gallon = 8 pints:
[tex]\[ 12.5 \text{ pints} \div 8 \text{ pints/gallon} = 1.5625 \text{ gallons} \][/tex]
or
[tex]\[ 1 \frac{9}{16} \text{ gallons} \][/tex]
#### 4. Convert 42 quarts to gallons.
Convert quarts to gallons, knowing that 1 gallon = 4 quarts:
[tex]\[ 42 \text{ quarts} \div 4 \text{ quarts/gallon} = 10.5 \text{ gallons} \][/tex]
### Matching the Converted Values to Given Equivalences
1. \( 16 \frac{1}{2} \) cups is equivalent to \( 132 \) fluid ounces.
2. 196 cups is equivalent to \( 12 \frac{1}{4} \) gallons.
3. \( 12 \frac{1}{2} \) pints is equivalent to \( 1.5625 \) or \( 1 \frac{9}{16} \) gallons (not matching exactly to the provided gallons since we round for simplicity).
4. 42 quarts is equivalent to \( 10 \frac{1}{2} \) gallons.
### Final Matching of Measurements
- \( 16 \frac{1}{2} \) cups corresponds to \( 132 \) fluid ounces.
- 196 cups corresponds to \( 12 \frac{1}{4} \) gallons.
- \( 12 \frac{1}{2} \) pints corresponds roughly to \( 1 \frac{9}{16} \) gallons.
- 42 quarts corresponds to \( 10 \frac{1}{2} \) gallons.
These paired combinations show which measurements are equivalent. Thus, drawing lines between them would result in:
- \( 16 \frac{1}{2} \) cups [tex]$\leftrightarrow$[/tex] \( 132 \) fluid ounces
- 196 cups [tex]$\leftrightarrow$[/tex] \( 12 \frac{1}{4} \) gallons
- \( 12 \frac{1}{2} \) pints [tex]$\leftrightarrow$[/tex] this converts to approximately \( 1.5625 \) gallons, closest to is given for understanding.
- 42 quarts [tex]$\leftrightarrow$[/tex] [tex]\( 10 \frac{1}{2} \)[/tex] gallons
### Conversion Steps and Equivalences
#### 1. Convert \( 16 \frac{1}{2} \) cups to fluid ounces.
First, convert the mixed number to an improper fraction or decimal:
[tex]\[ 16 \frac{1}{2} = 16 + \frac{1}{2} = 16.5 \text{ cups} \][/tex]
Next, convert cups to fluid ounces:
[tex]\[ 16.5 \text{ cups} \times 8 \text{ fl ounces/cup} = 132 \text{ fl ounces} \][/tex]
#### 2. Convert 196 cups to gallons.
Convert cups to gallons knowing that 1 gallon = 16 cups:
[tex]\[ 196 \text{ cups} \div 16 \text{ cups/gallon} = 12.25 \text{ gallons} \][/tex]
#### 3. Convert \( 12 \frac{1}{2} \) pints to gallons.
Convert the mixed number to an improper fraction or decimal:
[tex]\[ 12 \frac{1}{2} = 12.5 \text{ pints} \][/tex]
Next, convert pints to gallons, knowing that 1 gallon = 8 pints:
[tex]\[ 12.5 \text{ pints} \div 8 \text{ pints/gallon} = 1.5625 \text{ gallons} \][/tex]
or
[tex]\[ 1 \frac{9}{16} \text{ gallons} \][/tex]
#### 4. Convert 42 quarts to gallons.
Convert quarts to gallons, knowing that 1 gallon = 4 quarts:
[tex]\[ 42 \text{ quarts} \div 4 \text{ quarts/gallon} = 10.5 \text{ gallons} \][/tex]
### Matching the Converted Values to Given Equivalences
1. \( 16 \frac{1}{2} \) cups is equivalent to \( 132 \) fluid ounces.
2. 196 cups is equivalent to \( 12 \frac{1}{4} \) gallons.
3. \( 12 \frac{1}{2} \) pints is equivalent to \( 1.5625 \) or \( 1 \frac{9}{16} \) gallons (not matching exactly to the provided gallons since we round for simplicity).
4. 42 quarts is equivalent to \( 10 \frac{1}{2} \) gallons.
### Final Matching of Measurements
- \( 16 \frac{1}{2} \) cups corresponds to \( 132 \) fluid ounces.
- 196 cups corresponds to \( 12 \frac{1}{4} \) gallons.
- \( 12 \frac{1}{2} \) pints corresponds roughly to \( 1 \frac{9}{16} \) gallons.
- 42 quarts corresponds to \( 10 \frac{1}{2} \) gallons.
These paired combinations show which measurements are equivalent. Thus, drawing lines between them would result in:
- \( 16 \frac{1}{2} \) cups [tex]$\leftrightarrow$[/tex] \( 132 \) fluid ounces
- 196 cups [tex]$\leftrightarrow$[/tex] \( 12 \frac{1}{4} \) gallons
- \( 12 \frac{1}{2} \) pints [tex]$\leftrightarrow$[/tex] this converts to approximately \( 1.5625 \) gallons, closest to is given for understanding.
- 42 quarts [tex]$\leftrightarrow$[/tex] [tex]\( 10 \frac{1}{2} \)[/tex] gallons
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