Chapter 4 Represent And Solve Equations And Inequalities
Page 223 Exercise 1 Answer
In the video, we see someone trying to fit as many shoes as possible into a suitcase. Some possible questions that come to mind are then:
Why does he need to bring so many shoes?
How many shoes will fit inside the suitcase?
What will be the weight of the bag once his shoes are packed?
Result
Possible questions:
Why does he need to bring so many shoes?
How many shoes will fit inside the suitcase?
What will be the weight of the bag once his shoes are packed?
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 223 Exercise 2 Answer
In the video, we saw someone trying to fit as many shoes as possible into a suitcase. We were also given in the video the weights of the suitcase as he added each pair of shoes and we were told the weight limit was 50 pounds. The Main Question for the video is then:
How many shoes can he fit in his suitcase if the weight limit is 50 pounds?
Result
How many shoes can he fit in his suitcase if the weight limit is 50 pounds?
Page 223 Exercise 3 Answer
We need to make a prediction about how many shoes will fit. To make a prediction, we need to use an estimate for the weight of each pair of shoes and the weight of the suitcase.
Assuming the suitcase weighs 6 pounds, then the shoes will weigh 50 − 6 = 44 pounds. Assuming each pair of shoes weighs 2 pounds, then a prediction could be 22 pairs of shoes since 44 ÷ 2 = 22.
Result
Possible answer:
I predict that 22 pairs of shoes will fit by using an estimate of 6 pounds for the weight of the suitcase and 2 pounds for each pair of shoes.
Page 223 Exercise 4 Answer
We know the weight limit is 50 pounds.
A number that is too small to be the answer could then be 2, since this would mean 2 pairs of shoes and the suitcase would have a combined weight of 50 pounds, which is unrealistic.
A number that is too large to be the answer could be 100 since this would mean each pair of shoes weighed less than 50 ÷ 100 = 0.5 pounds, which is unrealistic.
Writing these two numbers on the given number line gives:
Result
Possible answers:
Too small: 2
Too large: 100
Page 223 Exercise 5 Answer
The prediction I made in Exercise 3 was 22 pounds. Plotting this on the number line from Exercise 4 then gives:
Result
Plot the prediction you made in Exercise 3 on the number line from Exercise 4.
Page 224 Exercise 6 Answer
To determine how many pairs of shoes he can fit, there are three pieces of information we need to know:
How much the suitcase weighs.
How much each pair of shoes weighs.
What the weight limit is.
We can then use this information to write an equation or an inequality that represents the total weight of the suitcase.
Result
We need to know how the suitcase weighs, how much each pair of shoes weighs, and what the weight limit is. we can use this information to write an equation or inequality that represents the total weight of the suitcase.
Page 224 Exercise 7 Answer
To get the information we need, we can use the information given by the Act 2 images in the online interactivity:
Weight of each pair of shoes: 2.2 lb, 2.5 lb, 2.2 lb, and 2.3 lb
Weight limit: 50 lb
Weight of suitcase: 6.2 lb
Result
Weight of each pair of shoes: 2.2 lb, 2.5 lb, 2.2 lb, and 2.3 lb
Weight limit: 50 lb
Weight of suitcase: 6.2 lb
Page 224 Exercise 8 Answer
To represent the situation, we need to use the following relationship:
weight of shoes + weight of suitcase ≤ weight limit
Possible solution:
We know that four pairs of shoes have weights of 2.2 lb, 2.5 lb, 2.2 lb, and 2.3 lb. To write an expression for the total weight of the shoes, we can use an estimate for the average weight of each pair of shoes. To find the estimated average, add the four weights and then divide by 4:
\(\frac{2.2+2.5+2.2+2.3}{4}=\frac{9.2}{4}=2.4\)The average weight of each pair is then 2.3 lb.
Let p be the number of pairs of shoes. Since the average weight of each pair is about 2.3 lb, then the total weight of p pairs of shoes is 2.3p pounds.
Since the suitcase weighs 6.2 lb, then the total weight of the suitcase and shoes is 2.3p + 6.2 pounds.
The weight limit is 50 lb so 2.3p + 6.2 must be less than or equal to 50. The inequality is then 2.3p + 6.2 ≤ 50.
Solving this inequality gives:
2.3p + 6.2 ≤ 50
2.3p + 6.2 − 6.2 ≤ 50 − 6.2 Subtract 6.2 on both sides.
2.3p ≤ 43.8 Simplify.
2.3p ÷ 2.3 ≤ 43.8 ÷ 2.3 Divide both sides by 2.3.
p ≤ 19.04 Simplify.
The number of pairs must be a whole number so if the number of pairs is at most 19.04, then the greatest number of pairs is 19 pairs.
Note: Since we only know the weights of 4 pairs of shoes, the actual average weight of the shoes may not be 2.3 lb. There could be pairs of shoes that weigh less than 2.2 lb or weigh more than 2.5 lb. This means that you can represent the situation using an average weight that is not 2.3.
For example, you could use an average of 2.4 lb or 2.5 lb since it is possible that the rest of the shoes have weights closer to 2.5 lb or you could use an average weight of 2.2 lb or 2.1 lb since it is possible that the rest of the shoes have weights closer to 2.2 lb.
Result
Possible answers: 2.3p + 6.2 ≤ 50 19 pairs
Page 224 Exercise 9 Answer
From Exercise 8, we got an answer of 19 pairs. In Exercise 3, I predicted 22 pairs. The answer of 19 pairs is then lower than my prediction. This is because I used an average weight of 2 lb for each pair of shoes in my prediction but an average of 2.3 lb was used in Exercise 8. A larger average means fewer pairs of shoes can fit.
Page 225 Exercise 10 Answer
From the Act 3 video, he had weights of 47.2 lb for 18 pairs of shoes, 49.3 for 19 pairs, and 51.9 lb for 20 pounds. Since the limit is 50 lb, then the answer to the Main Question is 19 pairs since this was the most pairs he could fit and still have a weight less than 50 lb.
Result
19 pairs
Page 225 Exercise 11 Answer
Possible solution: In Exercise 8, I used an average weight of 2.3 lb per pair of shoes to write the inequality 2.3p + 6.2 ≤ 50. This inequality gave me an answer of 19 pairs, which matches the answer in the video.
If you did not use the same inequality I did, your answer likely won’t match. If you got a higher answer, it’s because you used a lower average weight so the reason for the difference is that the shoes weighed more than you thought. Similarly, if you got a lower answer, it’s because you used a higher average weight so the reason for the difference is that the shoes weighed less than you thought.
Result
Possible answer: In Exercise 8, I used an average weight of 2.3 lb per pair of shoes to write the inequality 2.3p + 6.2 ≤ 50. This inequality gave me an answer of 19 pairs, which matches the answer in the video.
Page 225 Exercise 12 Answer
Possible answer: In Exercise 8, I used the inequality 2.3p + 6.2 ≤ 50, which gave me an answer of 19 pairs. Since my answer matches the video, I would not change my model.
If you did not use the same inequality I did, your answer likely didn’t match so you should change your model. If you got a higher answer, it’s because you used a lower average weight so you need to change your model to use a higher average weight. Similarly, if you got a lower answer, it’s because you used a higher average weight so you need to change your model to use a lower average weight.
Result
Possible answer: In Exercise 8, I used the inequality 2.3p + 6.2 ≤ 50, which gave me an answer of 19 pairs. Since my answer matches the video, I would not change my model.
Page 226 Exercise 13 Answer
Possible answer: I used a mathematical model to represent the situation by writing an inequality. I wrote the inequality using the following relationship:
weight of shoes + weight of suitcase ≤ weight limit
To write an expression for the weight of shoes, I let p be the number of pairs of shoes and multiplied p by an estimate for the average weight of the shoes. I then used this expression, the given weight of 6.2 lb for the suitcase, and the given weight limit of 50 lb to write the inequality. Solving this inequality for p and rounding to the nearest whole number then allowed me to answer the Main Question.
Result
I wrote the inequality weight of shoes + weight of suitcase ≤ weight limit to represent the situation. To write an expression for the weight of the shoes, I let p be the number of pairs of shoes and then multiplied p by an estimate for the average weight of the shoes. I then used this expression and the given weights for the suitcase and weight limit to write the inequality. Solving this inequality for p allowed me to answer the Main Question.
Page 226 Exercise 14 Answer
An inequality was more useful to answer the Main Question. This is because the combined weight of the shoes and suitcase did not have to equal 50 lb, it could also be less than 50 lb.
Page 226 Exercise 15 Answer
If the weight limit is changed to 40 pounds, then the number of shoes would decrease.
Changing the inequality 2.3p + 6.2 ≤ 50 from Exercise 8 to 2.3p + 6.2 ≤ 40 and solving for p gives:
2.3p + 6.2 − 6.2 ≤ 40 − 6.2 Subtract 6.2 on both sides.
2.3p ≤ 33.8 Simplify.
2.3p ÷ 2.3 ≤ 33.8 ÷ 2.3 Divide both sides by 2.3
p ≤ 14.70 Simplify.
The maximum number of shoes would then decrease to 14 pairs.
Result
If the weight limit is changed to 40 pounds, then the maximum number of shoes would decrease to 14 pairs.