- Calculate the unit price: \(\frac{\$1.95}{3 \text{ lbs}} = \$0.65/\text{lb}\).
- Formulate the cost equation: \(y = 0.65x\).
- Evaluate statement (a): \(\$0.65
eq \$0.75\), so it is false.
- Evaluate statements (b) and (c): \(y=0.65x\) is correct, and \(7 \times \$0.65 = \$4.55\), so both are true.
- Final answer: \(\boxed{\text{a}}\).
Explanation
- Problem Analysis
We are given that 3 pounds of bananas cost \(\$1.95\). Assuming a constant unit price, we can model this relationship as a linear equation of the form \(y = kx\), where \(y\) is the total cost in dollars, \(x\) is the weight in pounds, and \(k\) is the unit price (the cost per pound). Our objective is to determine the unit price and evaluate the truthfulness of three statements regarding the cost of bananas.
- Calculating the Unit Price
To find the unit price \(k\), we divide the total cost by the weight:
\[k = \frac{\$1.95}{3 \text{ lbs}} = \$0.65 \text{ per pound.}\]
Thus, the relationship between the weight in pounds (\(x\)) and the cost in dollars (\(y\)) is given by the linear equation:
\[y = 0.65x.\]
- Evaluating Statements and Conclusion
Now we evaluate each statement:
- Statement (a): "One pound of bananas is \(\$0.75\)." We calculated the unit price to be \(\$0.65\). Therefore, this statement is false.
- Statement (b): "The equation \(y = 0.65x\) could be used to determine the cost of \(x\) pounds of bananas." Our derived equation matches this, so this statement is true.
- Statement (c): "Seven pounds of bananas is \(\$4.55\)." Using our equation with \(x = 7\), we get \(y = 0.65 \times 7 = \$4.55\). This statement is also true.
Since we are looking for the statement that is not true, statement (a) is the correct choice.
Examples
Understanding unit rates is essential for smart shopping and budgeting. For instance, if you are buying fruit, paint, or fabric in bulk, calculating the cost per unit (like per pound or per gallon) allows you to compare different brands or package sizes to find the best value. By identifying the unit rate \(k\) in a linear relationship \(y = kx\), you can easily predict costs for any amount you might need to purchase.