Question

By using estimation techniques, determine which of the following is the heaviest and which is the lightest: a 5-lb bag of potatoes, a 5-kg bag of sugar, or \(1 \mathrm{gal}\) of water (density \(=1.0 \mathrm{~g} / \mathrm{mL}\) ).

Short answer

The heaviest item is the 5-kg bag of sugar with a weight of \(5000 \mathrm{~g}\), and the lightest item is the 5-lb bag of potatoes with a weight of \(2267.96 \mathrm{~g}\).

Step-by-step solution

Convert all weights to grams

We will first convert the weights of each item into grams. We have: 1. 5-lb bag of potatoes 2. 5-kg bag of sugar 3. 1 gallon of water (density \(=1.0 \mathrm{g/mL}\)) 1 lb = 453.592 g 1 kg = 1000 g 1 gallon = 3785.41 mL Now we convert each item's weight into grams: 1. 5-lb bag of potatoes: \(5 \times 453.592 = 2267.96 \mathrm{~g}\) 2. 5-kg bag of sugar: \(5 \times 1000 = 5000 \mathrm{~g}\) 3. 1 gallon of water: \(1 \times 3785.41 \mathrm{~mL} \times 1.0 \mathrm{~g/mL} = 3785.41 \mathrm{~g}\)

Compare the weights

Now we can compare the weights of the three items in grams: 1. 5-lb bag of potatoes: \(2267.96 \mathrm{~g}\) 2. 5-kg bag of sugar: \(5000 \mathrm{~g}\) 3. 1 gallon of water: \(3785.41 \mathrm{~g}\) Comparing these values, it is clear that: - The heaviest item is the 5-kg bag of sugar with a weight of \(5000 \mathrm{~g}\). - The lightest item is the 5-lb bag of potatoes with a weight of \(2267.96 \mathrm{~g}\).

Key concepts

Estimation Techniques

Estimating weights and conversions involves using approximate calculations to quickly gauge the weight of different items. This technique helps when you're trying to compare objects with different units of measurement. In our problem, we're comparing a 5-lb bag of potatoes, a 5-kg bag of sugar, and 1 gallon of water.

Rather than using precision measurement right away, we use estimation techniques by rounding conversion factors or simplifying calculations to get a rough idea. For example, knowing that 1 lb is approximately 450 g instead of 453.592 g can quickly help estimate without requiring a calculator.

Estimation is useful when:

• Quick decisions are needed.
• Exact precision is not crucial.
• Initial guesses before detailed work are valuable.

Density Calculations

Density is defined as mass per unit volume, represented by the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In our case, the density of water is given as 1.0 g/mL, which makes calculating the weight of 1 gallon of water straightforward.

To understand this better, if you know the volume of a liquid and its density, you can easily calculate its mass. Since 1 gallon equals 3785.41 mL for water, and knowing the density, the mass can be determined by multiplying the volume by the density, which provides 3785.41 g in this problem.

Density calculations help in:

• Transforming volume measurements to mass for liquids.
• Determining buoyancy or whether something will float.
• Comparing different materials through their density.

Unit Conversion

Unit conversion is the method of converting different units of measurement to one standard unit to make comparison easier. In scientific problems, this typically involves converting between metric and imperial systems.

For our problem, the weights of potatoes and water had to be converted to grams, similar to the sugar, which was already in kilograms. Knowing the following conversion factors helps:

• 1 lb is 453.592 g.
• 1 kg is 1000 g.
• 1 gallon is 3785.41 mL.

Implementing these conversions made it possible to easily compare the weights.

Benefits of unit conversion include:

• Eliminating confusion between measurement systems.
• Simplifying calculations and comparisons.
• Ensuring accuracy across different contexts.

Weight Comparison

After converting all measurements to the same unit, comparing weights becomes a matter of looking at the numbers. Our exercise demonstrated that once converted to grams, it was easy to adjudicate that the bag of sugar, weighing 5000 g, is the heaviest, and the potatoes, weighing 2267.96 g, are the lightest.

Making accurate comparisons requires ensuring that all items are in the same unit, avoiding mistakes from unequal units.

Weight comparison is crucial for:

• Understanding the mass differences between objects.
• Ranking items by weight for practical decisions.
• Ensuring equitable distribution in recipes or material use.

This basic concept underlies everything from cooking to shipping logistics.