Question

By using estimation technique, determine which of the following is the heaviest and which is the lightest: a \(10-1 \mathrm{~b}\) bag of fertilizer, a \(10-\mathrm{kg}\) bag of rice, or 2 gal of olive oil (density \(\left.=0.918 \mathrm{~g} / \mathrm{cm}^{3}\right)\)

Short answer

Based on the estimation technique, the heaviest item is the 10 kg bag of rice, weighing 10,000 g, and the lightest item is the 10-1 lb bag of fertilizer, weighing approximately 4,082.4 g.

Step-by-step solution

Convert the weights into a common unit (grams)

To compare the weights, we need to have them in a common unit. All three items have different units: pounds (lb), kilograms (kg), and gallons. We'll convert all the weights to grams (g).

Step 1a: Convert the bag of fertilizer weight to grams

The bag of fertilizer weighs 10-1 lb. First, we need to estimate 10 - 1 to obtain the approximate weight. 10-1 is close to 9, so we'll consider the bag of fertilizer to have an estimated weight of 9 lb. Next, we'll convert 9 lb to grams. We know that 1 lb is approximately 453.6 g. So, \( 9\ \text{lb} * 453.6\ \text{g/lb} \approx 4082.4\ \text{g} \)

Step 1b: Convert the bag of rice weight to grams

The bag of rice weighs 10 kg. Converting kilograms to grams, we have: \( 10\ \text{kg} * 1000\ \text{g/kg} = 10000\ \text{g} \)

Step 1c: Convert 2 gallons of olive oil weight to grams

First, we need to find the volume of olive oil in cubic centimeters (cm³). We know that 1 gallon is approximately 3,785.41 cm³. So, for 2 gallons, we have: \( 2\ \text{gal} * 3785.41\ \text{cm³/gal} \approx 7570.82\ \text{cm³} \) Now we can find the weight of olive oil in grams, using the given density (0.918 g/cm³). We multiply the volume by the density: \( 7570.82\ \text{cm³} * 0.918\ \text{g/cm³} \approx 6946.952\ \text{g} \)

Compare the weights

Now that we have the estimated weights of all three items in grams, we can compare them to determine which is the heaviest and which is the lightest. Fertilizer: \( \approx 4082.4\ \text{g} \) Rice: \( = 10000\ \text{g} \) Olive oil: \( \approx 6946.952\ \text{g} \) Comparing the estimated weights, we can conclude that the heaviest is the bag of rice (10,000 g), and the lightest is the bag of fertilizer (approx. 4,082.4 g).

Key concepts

Density

Density is a key concept when it comes to understanding the weight and volume of a material. Simply put, density measures how much mass is contained in a given volume. The formula for density is \( \text{density} = \frac{\text{mass}}{\text{volume}} \). In our exercise, the density of olive oil is given as 0.918 g/cm³. This means that each cubic centimeter (cm³) of olive oil has a mass of 0.918 grams.

Knowing the density of a substance allows us to convert its volume to mass and vice versa. For example, when you have 2 gallons of olive oil, you can convert the gallons to cubic centimeters and then use the density to find out how much it weighs in grams. This conversion is vital for comparing weights when the items are not initially in the same units.

It's important to remember that liquids often have their densities measured in g/cm³, while solids can vary depending on the material. Understanding density helps in situations where you need to compare different materials or estimate the outcome when only volume information is available.

Weight Comparison

When comparing the weights of various items, it's essential to bring them into a common unit. In our problem, we converted all items to grams to make a straightforward comparison possible. This way, we avoid confusion caused by different measurement units like pounds, kilograms, or gallons.

Once everything is measured in the same unit, you can easily line up the weights side-by-side: 10 kg of rice converted to 10,000 g, 2 gallons of olive oil to approximately 6,947 g, and a 10-1 lb bag of fertilizer to about 4,082 g under estimation. These conversions revealed that the rice is the heaviest, followed by the olive oil, and finally the fertilizer bag is the lightest.

Weight comparison is a common requirement in both daily tasks and scientific calculations. Being able to switch between units and having a reliable point of reference (such as grams) makes the process efficient and clear.

Estimation Technique

Estimation is a handy technique employed to simplify calculations when dealing with complex numbers or operations in real-world situations. It allows us to make quick decisions based on approximate values rather than precise numbers. In our case, to estimate the weight of a fertilizer bag described as "10-1 lb," we approximated it to 9 lb before converting it to grams.

Estimation is especially valuable in scenarios where precise values are not readily available or when you need a quick solution. It involves rounding numbers to more manageable figures that are close enough to give us a good idea of the actual value. This might include rounding off decimal points and simplifying fractions.

Good estimation skills come in handy with mental math, help in checking the plausibility of results, and offer a fast way to communicate figures. While estimation does not provide the exact answer, it is an excellent tool to set the ballpark figure, guiding further detailed calculations or decisions.