For the roof of the shelter, Mia has a board that measures 1 square yard, a second
board that measures 3 feet by 3 feet, and a third board that measures 36 inches by
36 inches.
Which of the boards, if any, will cover all the space needed for the roof? Explain your
answer, drawing a picture if necessary.
1 foot = 1/3 of a yard
make it easy
Let's compare the sizes of the three boards Mia has to cover the roof of the shelter.
1. The first board measures 1 square yard.
2. The second board measures 3 feet by 3 feet.
3. The third board measures 36 inches by 36 inches.
To easily compare these measurements, we can convert them all into the same unit. Given that 1 yard equals 3 feet, and 1 foot equals 12 inches, we can make the following conversions:
1 square yard is equivalent to:
- 1 yard * 3 feet/yard = 3 feet on one side
- So, 3 feet * 3 feet = 9 square feet
The second board measures 3 feet by 3 feet, which is:
- 3 feet * 3 feet = 9 square feet (Same as above.)
The third board measures 36 inches by 36 inches. Since there are 12 inches in a foot, the board in feet is:
- 36 inches / 12 inches/foot = 3 feet on one side
- So, 3 feet * 3 feet = 9 square feet (Again, the same as above.)
After converting, we see that all three boards are indeed the same size when their dimensions are converted into the same units. Each board measures 3 feet by 3 feet, corresponding to 9 square feet, or 1 square yard when heights and widths are considered.
Thus, any of the boards will cover the same area of space needed for the roof, and it comes down to Mia's preference for which material to use based on other factors, such as thickness, weight, or ease of use.
make it shorter
All three boards that Mia has are equivalent in size because they all measure 3 feet by 3 feet, which is also the same as 1 square yard or 36 inches by 36 inches. Converting all the dimensions to the same unit, we find that each board will cover 9 square feet (or 1 square yard) of space. Therefore, any of the boards can be used to cover the space needed for the roof.
shorter
All three boards are the same size: 1 square yard, 3 feet by 3 feet, and 36 inches by 36 inches are different ways to describe the same area. Any board will cover the roof space needed, as they are all equivalent in size.
Roger is planning a party. He needs these ingredients in the following quantities to
make punch:
• 2 quarts orange juice • 3 cups cranberry juice • 1 gallon ginger ale
• 1 pint pineapple juice
Roger knows he will need a relatively large container to mix the punch. Unfortunately,
all he can find is a bowl that holds 8 liters of liquid. He is not sure whether he can fit
all of the ingredients in the bowl at the same time.
Use the following measurement equivalents to determine if Roger can use the 8-liter
bowl to mix all the ingredients for the punch. 1 liter is about 4.23 cups, 2.11 pints,
1.06 quarts, and 0.26 gallons. Show your work.
First, let's convert all the ingredients Roger needs for the punch into one common unit. We'll choose liters for consistency, and we'll use the given measurement equivalents for the conversion.
Ingredients:
1. 2 quarts of orange juice
2. 3 cups of cranberry juice
3. 1 gallon of ginger ale
4. 1 pint of pineapple juice
Conversions (using the given equivalents):
- 1 quart is approximately 1.06 liters.
- 1 cup is approximately 1/4.23 liters.
- 1 gallon is approximately 3.785 liters (since 1 gallon = 4 quarts and 1 quart ≈ 1.06 liters).
- 1 pint is approximately 2.11 pints per liter.
Now let's calculate:
1. Orange juice: 2 quarts × 1.06 liters/quart = 2.12 liters.
2. Cranberry juice: 3 cups × (1/4.23 liters/cup) ≈ 3 / 4.23 liters ≈ 0.71 liters.
3. Ginger ale: 1 gallon × 3.785 liters/gallon = 3.785 liters.
4. Pineapple juice: 1 pint × (1/2.11 pints/liter) = 1 / 2.11 liters ≈ 0.47 liters.
Finally, let’s add up all the ingredients in liters:
2.12 liters (orange juice) + 0.71 liters (cranberry juice) + 3.785 liters (ginger ale) + 0.47 liters (pineapple juice) = 7.085 liters total.
Since 7.085 liters is less than the 8 liters capacity of the bowl, Roger can indeed use the bowl to mix all of the ingredients for the punch at the same time.
If Roger can use the bowl, he needs to determine how many liters of liquid could still
fit into the bowl. If he cannot use the bowl, he needs to determine how many extra
liters of liquid there are. Round your answers to the nearest tenth of a liter. Show
your work.
In the previous explanation, we calculated the total volume of the punch ingredients in liters as 7.085 liters. Roger has a bowl that holds 8 liters. Now we'll determine how much more liquid can fit into the bowl by subtracting the total volume of the punch ingredients from the bowl's capacity.
Bowl capacity - Total volume of ingredients = Remaining volume
8 liters (bowl capacity) - 7.085 liters (total punch ingredients) ≈ 0.915 liters
Now we'll round this to the nearest tenth of a liter:
0.915 liters ≈ 0.9 liters
So, after combining all the punch ingredients, the bowl will still be able to hold about 0.9 liters of additional liquid.
Roger also wants to make sandwiches for his guests. He buys the following sandwich
fillings:
• 1.5 kilograms of turkey
• 1.25 kilograms of beef
• 2.25 pounds of cheese
a. Roger wants to know if he purchased more turkey or more cheese. Use the
conversion ratio 1 kilogram = about 2.2 pounds to determine the correct
answer. Show your work.
First, we'll convert the weight of the cheese from pounds to kilograms using the conversion ratio 1 kilogram ≈ 2.2 pounds.
The weight of the cheese is:
2.25 pounds ÷ 2.2 pounds/kilogram ≈ 1.023 kilograms
Now, we have the weights in the same unit:
- Turkey: 1.5 kilograms
- Cheese: approximately 1.023 kilograms
Comparing these weights:
- Turkey: 1.5 kilograms
- Cheese: ~1.023 kilograms
It's clear that Roger purchased more turkey (1.5 kilograms) than cheese (~1.023 kilograms).
Roger discovers that he needs to buy some chicken as well. He buys an
amount of chicken that weighs more than the beef he bought and less than the
turkey he bought. Give an amount in pounds that the chicken could weigh,
rounded to the nearest hundredth of a pound. Show your work.
Roger bought 1.25 kilograms of beef and 1.5 kilograms of turkey. To determine a possible weight for the chicken in pounds that is between these two amounts, we first need to convert both the turkey and beef weights into pounds using the conversion ratio 1 kilogram ≈ 2.2 pounds.
Beef weight in pounds:
1.25 kg × 2.2 pounds/kg ≈ 2.75 pounds
Turkey weight in pounds:
1.5 kg × 2.2 pounds/kg ≈ 3.3 pounds
Now we need to choose a weight for the chicken that is more than 2.75 pounds and less than 3.3 pounds. Let's select a number that is between these two values. For example, we could say the chicken weighs 3.0 pounds.
Therefore, the chicken could weigh 3.00 pounds (rounded to the nearest hundredth of a pound), which is greater than the beef's weight of 2.75 pounds and less than the turkey's weight of 3.3 pounds.
1.5kg * 2.2 ound/kg
To convert 1.5 kilograms to pounds using the conversion ratio 1 kilogram ≈ 2.2 pounds, you simply multiply the number of kilograms by the conversion factor:
1.5 kg * 2.2 pounds/kg = 3.3 pounds
So, 1.5 kilograms is equal to 3.3 pounds.
c. Roger is curious to know the total weight of all four sandwich fillings he bought.
He weighs the fillings twice, once using ounces and once using grams. If 16
ounces = 1 pound and 1,000 grams = 1 kilogram, will the total weight of the
sandwich fillings be a greater number of ounces or a greater number of
kilograms? Explain your answer.
Let's first find the total weight in kilograms by adding the weight of the turkey, beef, and the estimated weight of the chicken (using a mid-range value between the weight of the beef and turkey in kilograms, as we do not have the exact number). We will exclude the cheese here since we're comparing the total in two different units (grams and ounces), not the total mass.
Total weight in kilograms:
Turkey: 1.5 kg
Beef: 1.25 kg
Estimated Chicken: Let's take the midpoint between the beef and turkey, which is about (1.5 kg + 1.25 kg)/2 = 1.375 kg.
Now, we add them together:
1.5 kg + 1.25 kg + 1.375 kg = 4.125 kg
Converting kilograms to grams (since 1,000 grams = 1 kilogram):
4.125 kg * 1,000 grams/kg = 4,125 grams
Now, we find the total weight in ounces by converting the kilograms to pounds and then pounds to ounces.
Total weight in pounds (using 1 kilogram ≈ 2.2 pounds):
Turkey: 1.5 kg * 2.2 lb/kg = 3.3 lb
Beef: 1.25 kg * 2.2 lb/kg = 2.75 lb
Estimated Chicken: 1.375 kg * 2.2 lb/kg = 3.025 lb
Adding these together gives us:
3.3 lb + 2.75 lb + 3.025 lb = 9.075 lb
Converting pounds to ounces (since 16 ounces = 1 pound):
9.075 lb * 16 oz/lb = 145.2 oz
Comparing the results:
- In grams, the total weight is 4,125 grams.
- In ounces, the total weight is 145.2 ounces.
Since 1 ounce is approximately 28.35 grams (and without converting the two, we can tell by just comparing the sizes of the numbers):
145.2 oz * 28.35 g/oz ≈ 4116.32 g
The total weight is a greater number of grams rather than ounces (4,125 grams is less than 4,116.32 grams). This indicates a logical discrepancy, as we should expect the comparison to reflect the same physical quantity regardless of the unit of measurement.
It's important to note that estimating the weight of the chicken (by taking an arbitrary midpoint) has introduced uncertainty into the calculations. To accurately conclude which number is greater, we need the exact weight of the chicken. However, in this context, assuming equal conversions in units of mass (grams to kilograms, ounces to pounds), one would not expect a difference between scales since the conversion rates should preserve the equivalence of the physical quantity represented by different units. The total weight should be the same physical quantity, just represented by different numerical values in different units.