Engage NY Eureka Math 4th Grade Module 5 Lesson 41 Answer Key

Eureka Math Grade 4 Module 5 Lesson 41 Problem Set Answer Key

Question 1.
Find the sums.
a. $$\frac{0}{3}+\frac{1}{3}+\frac{2}{3}+\frac{3}{3}$$

0/3 + 1/3 + 2/3 + 3/3 = 1.9.

Explanation:
In the above-given question,
given that,
Find the sums.
0/3 + 1/3 + 2/3 + 3/3.
0/3 = 0.
1/3 = 0.3.
2/3 = 0.6.
3/3 = 1.
0 + 0.3 + 0.6 + 1 = 1.9.

b. $$\frac{0}{4}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\frac{4}{4}$$

0/4 + 1/4 + 2/4 + 3/4 + 4/4 = 2.5.

Explanation:
In the above-given question,
given that,
Find the sums.
0/4 + 1/4 + 2/4 + 3/4 + 4/4.
0/4 = 0.
1/4 = 0.25.
2/4 = 0.5.
3/4 = 0.75
4/4 = 1.
0 + 0.25 + 0.5 + 0.75 + 1 = 2.5.

c. $$\frac{0}{5}+\frac{1}{5}+\frac{2}{5}+\frac{3}{5}+\frac{4}{5}+\frac{5}{5}$$

0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 = 3.

Explanation:
In the above-given question,
given that,
Find the sums.
0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5.
0/5 = 0.
1/5 = 0.2.
2/5 = 0.4.
3/5 = 0.6.
4/5 = 0.8.
5/5 = 1.
0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 = 3.

d. $$\frac{0}{6}+\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}+\frac{6}{6}$$

0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 3.3.

Explanation:
In the above-given question,
given that,
Find the sums.
0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6.
0/6 = 0.
1/6 = 0.1.
2/6 = 0.3.
3/6 = 0.5.
4/6 = 0.6.
5/6 = 0.8.
6/6 = 1
0 + 0.1 + 0.3 + 0.5 + 0.6 + 0.8 + 1 = 3.3.

e. $$\frac{0}{7}+\frac{1}{7}+\frac{2}{7}+\frac{3}{7}+\frac{4}{7}+\frac{5}{7}+\frac{6}{7}+\frac{7}{7}$$

0/7 + 1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7Â  + 7/7 = 3.7.

Explanation:
In the above-given question,
given that,
Find the sums.
0/7 + 1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7 + 7/7.
0/7 = 0.
1/7 = 0.1.
2/7 = 0.2.
3/7 = 0.4.
4/7 = 0.5.
5/7 = 0.7.
6/7 = 0.8.
7/7 = 1.
0 + 0.1 + 0.3Â  + 0.4 + 0.5 + 0.6 + 0.8 + 1 = 3.7.

f. $$\frac{0}{8}+\frac{1}{8}+\frac{2}{8}+\frac{3}{8}+\frac{4}{8}+\frac{5}{8}+\frac{6}{8}+\frac{7}{8}+\frac{8}{8}$$

0/8 + 1/8 + 2/8 + 3/8 + 4/8 + 5/8 + 6/8Â  + 7/8 = 4.2.

Explanation:
In the above-given question,
given that,
Find the sums.
0/8 + 1/8 + 2/8 + 3/8 + 4/8 + 5/8 + 6/8 + 7/8 + 8/8.
0/8 = 0.
1/8 = 0.125.
2/8 = 0.25.
3/8 = 0.37.
4/8 = 0.5.
5/8 = 0.6.
6/8 = 0.7.
7/8 = 0.8.
8/8 = 1.
0 + 0.1 + 0.3Â  + 0.4 + 0.5 + 0.6 + 0.8 + 1 = 4.2.

Question 2.
Describe a pattern you notice when adding the sums of fractions with even denominators as opposed to those with odd denominators.

The sum of fractions with even denominators increases.
the sum of fractions with odd denominators decreases.

Explanation:
In the above-given question,
given that,
when adding the sums of fractions with even denominators increases.
when adding the sums of fractions with odd denominators decreases.
0/3 = 1.9.
0/4 = 2.5.
0/5 = 3.
0/6 = 3.3.
0/7 = 3.7.
0/8 = 4.2.

Question 3.
How would the sums change if the addition started with the unit fraction rather than with 0?

The sum does not change.

Explanation:
In the above-given question,
given that,
How would the sums change if the addition started with the unit fraction?
0 plus anything is anything.
0 + 1 = 1.
so the sum does not change.

Question 4.
Find the sums.
a. $$\frac{0}{10}+\frac{1}{10}+\frac{2}{10}+\cdots+\frac{10}{10}$$

0/10 + 1/10 + 2/10 + 10/10 = 1.3.

Explanation:
In the above-given question,
given that,
Find the sums.
0/10 + 1/10 + 2/10 + 10/10.
0/10 = 0.
1/10 = 0.1.
2/10 = 0.2.
10/10 = 1.
0 + 0.1 + 0.2 + 1 = 1.3.

b. $$\frac{0}{12}+\frac{1}{12}+\frac{2}{12}+\cdots+\frac{12}{12}$$

0/12 + 1/12 + 2/12 + 12/12 = 1.18.

Explanation:
In the above-given question,
given that,
Find the sums.
0/12 + 1/12 + 2/12 + 12/12.
0/12 = 0.
1/12 = 0.08.
2/12 = 0.1.
12/12 = 1.
0 + 0.08 + 0.1 + 1 = 1.18.

c. $$\frac{0}{15}+\frac{1}{15}+\frac{2}{15}+\cdots+\frac{15}{15}$$

0/15 + 1/15 + 2/15 + 15/15 = 1.16.

Explanation:
In the above-given question,
given that,
Find the sums.
0/15 + 1/15 + 2/15 + 15/15.
0/15 = 0.
1/15 = 0.06.
2/15 = 0.1.
15/15 = 1.
0 + 0.06 + 0.1 + 1 = 1.16.

d. $$\frac{0}{25}+\frac{1}{25}+\frac{2}{25}+\cdots+\frac{25}{25}$$

0/25 + 1/25 + 2/25 + 25/25 = 1.12.

Explanation:
In the above-given question,
given that,
Find the sums.
0/25 + 1/25 + 2/25 + 25/25.
0/25 = 0.
1/25 = 0.04.
2/25 = 0.08.
25/25 = 1.
0 + 0.04 + 0.08 + 1 = 1.12.

e. $$\frac{0}{50}+\frac{1}{50}+\frac{2}{50}+\cdots+\frac{50}{50}$$

0/50 + 1/50 + 2/50 + 50/50 = 1.06.

Explanation:
In the above-given question,
given that,
Find the sums.
0/50 + 1/50 + 2/50 + 50/50.
0/50 = 0.
1/50 = 0.02.
2/50 = 0.04.
50/50 = 1.
0 + 0.02 + 0.04 + 1 = 1.06.

f. $$\frac{0}{100}+\frac{1}{100}+\frac{2}{100}+\cdots+\frac{100}{100}$$

0/100 + 1/100 + 2/100 + 100/100 = 1.03.

Explanation:
In the above-given question,
given that,
Find the sums.
0/100 + 1/100 + 2/100 + 100/100.
0/100 = 0.
1/100 = 0.01.
2/100 = 0.02.
100/100 = 1.
0 + 0.01 + 0.02 + 1 = 1.03.

Question 5.
Compare your strategy for finding the sums in Problems 4(d), 4(e), and 4(f) with a partner.

The sum is the same.

Explanation:
In the above-given question,
given that,
my partner found also the same.
the sum is the same.

Question 6.
How can you apply this strategy to find the sum of all the whole numbers from 0 to 100?

The sum of all the whole numbers is the same.

Explanation:
In the above-given question,
given that,
we can find the sum of all the whole numbers from 0 to 100.
if we find the whole numbers from 0 to 100.
the sum increases.

Eureka Math Grade 4 Module 5 Lesson 41 Exit Ticket Answer Key

Find the sums.
Question 1.
$$\frac{0}{20}+\frac{1}{20}+\frac{2}{20}+\cdots+\frac{20}{20}$$

0/20 + 1/20 + 2/20 + 20/20 = 1.15.

Explanation:
In the above-given question,
given that,
Find the sums.
0/20 + 1/20 + 2/20 + 20/20.
0/20 = 0.
1/20 = 0.05.
2/20 = 0.1.
20/20 = 1.
0 + 0.05 + 0.1 + 1 = 1.15.

Question 2.
$$\frac{0}{200}+\frac{1}{200}+\frac{2}{200}+\cdots+\frac{200}{200}$$

0/200 + 1/200 + 2/200 + 200/200 = 1.015.

Explanation:
In the above-given question,
given that,
Find the sums.
0/200 + 1/200 + 2/200 + 200/200.
0/200 = 0.
1/200 = 0.005.
2/200 = 0.01.
200/200 = 1.
0 + 0.005 + 0.01 + 1 = 1.015.

Question 1.
Find the sums.
a. $$\frac{0}{5}+\frac{1}{5}+\frac{2}{5}+\frac{3}{5}+\frac{4}{5}+\frac{5}{5}$$

0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 = 3.

Explanation:
In the above-given question,
given that,
Find the sums.
0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5.
0/5 = 0.
1/5 = 0.2.
2/5 = 0.4.
3/5 = 0.6.
4/5 = 0.8.
5/5 = 1.
0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 = 3.

b. $$\frac{0}{6}+\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}+\frac{6}{6}$$

0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 3.3.

Explanation:
In the above-given question,
given that,
Find the sums.
0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6.
0/6 = 0.
1/6 = 0.1.
2/6 = 0.3.
3/6 = 0.5.
4/6 = 0.6.
5/6 = 0.8.
6/6 = 1
0 + 0.1 + 0.3 + 0.5 + 0.6 + 0.8 + 1 = 3.3.

c. $$\frac{0}{7}+\frac{1}{7}+\frac{2}{7}+\frac{3}{7}+\frac{4}{7}+\frac{5}{7}+\frac{6}{7}+\frac{7}{7}$$

0/7 + 1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7Â  + 7/7 = 3.7.

Explanation:
In the above-given question,
given that,
Find the sums.
0/7 + 1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7 + 7/7.
0/7 = 0.
1/7 = 0.1.
2/7 = 0.2.
3/7 = 0.4.
4/7 = 0.5.
5/7 = 0.7.
6/7 = 0.8.
7/7 = 1.

d. $$\frac{0}{8}+\frac{1}{8}+\frac{2}{8}+\frac{3}{8}+\frac{4}{8}+\frac{5}{8}+\frac{6}{8}+\frac{7}{8}+\frac{8}{8}$$

0/8 + 1/8 + 2/8 + 3/8 + 4/8 + 5/8 + 6/8Â  + 7/8 = 4.2.

Explanation:
In the above-given question,
given that,
Find the sums.
0/8 + 1/8 + 2/8 + 3/8 + 4/8 + 5/8 + 6/8 + 7/8 + 8/8.
0/8 = 0.
1/8 = 0.125.
2/8 = 0.25.
3/8 = 0.37.
4/8 = 0.5.
5/8 = 0.6.
6/8 = 0.7.
7/8 = 0.8.
8/8 = 1.
0 + 0.1 + 0.3Â  + 0.4 + 0.5 + 0.6 + 0.8 + 1 = 4.2.

e. $$\frac{0}{9}+\frac{1}{9}+\frac{2}{9}+\frac{3}{9}+\frac{4}{9}+\frac{5}{9}+\frac{6}{9}+\frac{7}{9}+\frac{8}{9}+\frac{9}{9}$$

0/9 + 1/9 + 2/9 + 3/9 + 4/9 + 5/9 + 6/9Â  + 7/9 + 8/9 + 9/9 = 4.6.

Explanation:
In the above-given question,
given that,
Find the sums.
0/9 + 1/9 + 2/9 + 3/9 + 4/9 + 5/9 + 6/9 + 7/9 + 8/9 + 9/9.
0/9 = 0.
1/9 = 0.1.
2/9 = 0.2.
3/9 = 0.3.
4/9 = 0.4.
5/9 = 0.5.
6/9 = 0.6.
7/9 = 0.7.
8/9 = 0.8.
9/9 = 1
0 + 0.1 + 0.2 + 0.3Â  + 0.4 + 0.5 + 0.6Â  + 0.7+ 0.8 + 1 = 4.6.

f. $$\frac{0}{10}+\frac{1}{10}+\frac{2}{10}+\frac{3}{10}+\frac{4}{10}+\frac{5}{10}+\frac{6}{10}+\frac{7}{10}+\frac{8}{10}+\frac{9}{10}+\frac{10}{10}$$

0/10 + 1/10 + 2/10 + 3/10 + 4/10 + 5/10 + 6/10Â  + 7/10 + 8/10 + 9/10 + 10/10 = 5.5.

Explanation:
In the above-given question,
given that,
Find the sums.
0/10 + 1/10 + 2/10 + 3/10 + 4/10 + 5/10 + 6/10 + 7/10 + 8/10 + 9/10 + 10/10.
0/10 = 0.
1/10 = 0.1.
2/10 = 0.2.
3/10 = 0.3.
4/10 = 0.4.
5/10 = 0.5.
6/10 = 0.6.
7/10 = 0.7.
8/10 = 0.8.
9/10 = 0.9.
10/10 = 1.
0 + 0.1 + 0.2 + 0.3Â  + 0.4 + 0.5 + 0.6Â  + 0.7+ 0.8Â  + 0.9 + 1 = 5.5.

Question 2.
Describe a pattern you notice when adding the sums of fractions with even denominators as opposed to those with odd denominators.

The sum of fractions with even denominators increases.
the sum of fractions with odd denominators decreases.

Explanation:
In the above-given question,
given that,
when adding the sums of fractions with even denominators increases.
when adding the sums of fractions with odd denominators decreases.
0/3 = 1.9.
0/4 = 2.5.
0/5 = 3.
0/6 = 3.3.
0/7 = 3.7.
0/8 = 4.2.

Question 3.
How would the sums change if the addition started with the unit fraction rather than with 0?

The sum does not change.

Explanation:
In the above-given question,
given that,
How would the sums change if the addition started with the unit fraction?
0 plus anything is anything.
0 + 1 = 1.
so the sum does not change.

Question 4.
Find the sums.
a. $$\frac{0}{20}+\frac{1}{20}+\frac{2}{20}+\cdots+\frac{20}{20}$$

0/20 + 1/20 + 2/20 + 20/20 = 1.15.

Explanation:
In the above-given question,
given that,
Find the sums.
0/20 + 1/20 + 2/20 + 20/20.
0/20 = 0.
1/20 = 0.05.
2/20 = 0.1.
20/20 = 1.
0 + 0.05 + 0.1 + 1 = 1.15.

b. $$\frac{0}{35}+\frac{1}{35}+\frac{2}{35}+\cdots+\frac{35}{35}$$

0/35 + 1/35 + 2/35 + 35/35 = 1.07.

Explanation:
In the above-given question,
given that,
Find the sums.
0/35 + 1/35 + 2/35 + 35/35.
0/35 = 0.
1/35 = 0.02.
2/35 = 0.05.
35/35 = 1.
0 + 0.02 + 0.05 + 1 = 1.07.

c. $$\frac{0}{36}+\frac{1}{36}+\frac{2}{36}+\cdots+\frac{36}{36}$$

0/36 + 1/36 + 2/36 + 36/36 = 1.25.

Explanation:
In the above-given question,
given that,
Find the sums.
0/36 + 1/36 + 2/36 + 36/36.
0/36 = 0.
1/36 = 1.07.
2/36 = 0.2.
36/36 = 0.05.
0 + 0.05 + 0.2 + 1 = 1.25.

d. $$\frac{0}{75}+\frac{1}{75}+\frac{2}{75}+\cdots+\frac{75}{75}$$

0/75 + 1/75 + 2/75 + 75/75 = 1.03.

Explanation:
In the above-given question,
given that,
Find the sums.
0/75 + 1/75 + 2/75 + 75/75.
0/75 = 0.
1/75 = 0.01.
2/75 = 0.02.
75/75 = 1.
0 + 0.01 + 0.02 + 1 = 1.03.

e. $$\frac{0}{100}+\frac{1}{100}+\frac{2}{100}+\cdots+\frac{100}{100}$$

0/100 + 1/100 + 2/100 + 100/100 = 1.03.

Explanation:
In the above-given question,
given that,
Find the sums.
0/100 + 1/100+ 2/100 + 100/100.
0/100 = 0.
1/100 = 0.01.
2/100 = 0.02.
100/100 = 1.
0 + 0.01 + 0.02 + 1 = 1.03.

f. $$\frac{0}{99}+\frac{1}{99}+\frac{2}{99}+\cdots+\frac{99}{99}$$

0/99 + 1/99 + 2/99 + 99/99 = 1.03.

Explanation:
In the above-given question,
given that,
Find the sums.
0/99 + 1/99 + 2/99 + 99/99.
0/99 = 0.
1/99 = 0.01.
2/99 = 0.02.
99/99 = 1.
0 + 0.01 + 0.02 + 1 = 1.03.

Question 5.
How can you apply this strategy to find the sum of all the whole numbers from 0 to 50? To 99?