Practice · Find patterns in mixed-number sequences and sum · Sensim Math

Variant 1 easy answer:

18\frac{2}{9}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{9},\ 3\tfrac{4}{9},\ 5\tfrac{6}{9},\ \ldots,\ 7\tfrac{8}{9}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{9}$ , 3 $\frac{4}{9}$ , 5 $\frac{6}{9}$ , ..., up to 7 $\frac{8}{9}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 9.
The last listed term is 7 $\frac{8}{9}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 7 $\frac{8}{9}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7 (odd numbers) and the numerators go 2, 4, 6, 8 (even numbers), all over 9. So there are 4 terms.

1\tfrac{2}{9},\,3\tfrac{4}{9},\,5\tfrac{6}{9},\,7\tfrac{8}{9}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 = 16.

1 + 3 + 5 + 7 = 16

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 9):

\frac{2 + 4 + 6 + 8}{9}

=

\frac{20}{9}

. Since

\frac{20}{9}

= 2 with remainder 2, this is 2

\frac{2}{9}

.

\dfrac{2 + 4 + 6 + 8}{9}=\dfrac{20}{9}=2\tfrac{2}{9}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 16 + 2

\frac{2}{9}

= 18

\frac{2}{9}

.

16+2\tfrac{2}{9}=18\tfrac{2}{9}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

18\frac{2}{9}

4 · Reviewdoes it hold up?

There are 4 terms; the whole parts alone sum to 16, and the fraction parts add only 2 $\frac{2}{9}$ more, landing on 18 $\frac{2}{9}$ . The final fraction part $\frac{2}{9}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 18 $\frac{2}{9}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 16.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{20}{9}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 9ths!

Variant 2 easy answer:

18

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{10},\ 3\tfrac{4}{10},\ 5\tfrac{6}{10},\ \ldots,\ 7\tfrac{8}{10}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{10}$ , 3 $\frac{4}{10}$ , 5 $\frac{6}{10}$ , ..., up to 7 $\frac{8}{10}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 10.
The last listed term is 7 $\frac{8}{10}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 7 $\frac{8}{10}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7 (odd numbers) and the numerators go 2, 4, 6, 8 (even numbers), all over 10. So there are 4 terms.

1\tfrac{2}{10},\,3\tfrac{4}{10},\,5\tfrac{6}{10},\,7\tfrac{8}{10}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 = 16.

1 + 3 + 5 + 7 = 16

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 10):

\frac{2 + 4 + 6 + 8}{10}

=

\frac{20}{10}

. Since

\frac{20}{10}

= 2 with remainder 0, this is 2.

\dfrac{2 + 4 + 6 + 8}{10}=\dfrac{20}{10}=2

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 16 + 2 = 18.

16+2=18

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

18

4 · Reviewdoes it hold up?

There are 4 terms; the whole parts alone sum to 16, and the fraction parts add only 2 more, landing on 18. The final fraction part $\frac{0}{10}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 18.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 16.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{20}{10}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 10ths!

Variant 3 easy answer:

27\frac{8}{11}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{11},\ 3\tfrac{4}{11},\ 5\tfrac{6}{11},\ \ldots,\ 9\tfrac{10}{11}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{11}$ , 3 $\frac{4}{11}$ , 5 $\frac{6}{11}$ , ..., up to 9 $\frac{10}{11}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 11.
The last listed term is 9 $\frac{10}{11}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 9 $\frac{10}{11}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9 (odd numbers) and the numerators go 2, 4, 6, 8, 10 (even numbers), all over 11. So there are 5 terms.

1\tfrac{2}{11},\,3\tfrac{4}{11},\,5\tfrac{6}{11},\,7\tfrac{8}{11},\,9\tfrac{10}{11}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 = 25.

1 + 3 + 5 + 7 + 9 = 25

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 11):

\frac{2 + 4 + 6 + 8 + 10}{11}

=

\frac{30}{11}

. Since

\frac{30}{11}

= 2 with remainder 8, this is 2

\frac{8}{11}

.

\dfrac{2 + 4 + 6 + 8 + 10}{11}=\dfrac{30}{11}=2\tfrac{8}{11}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 25 + 2

\frac{8}{11}

= 27

\frac{8}{11}

.

25+2\tfrac{8}{11}=27\tfrac{8}{11}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

27\frac{8}{11}

4 · Reviewdoes it hold up?

There are 5 terms; the whole parts alone sum to 25, and the fraction parts add only 2 $\frac{8}{11}$ more, landing on 27 $\frac{8}{11}$ . The final fraction part $\frac{8}{11}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 27 $\frac{8}{11}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 25.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{30}{11}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 11ths!

Variant 4 medium answer:

27\frac{1}{2}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{12},\ 3\tfrac{4}{12},\ 5\tfrac{6}{12},\ \ldots,\ 9\tfrac{10}{12}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{12}$ , 3 $\frac{4}{12}$ , 5 $\frac{6}{12}$ , ..., up to 9 $\frac{10}{12}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 12.
The last listed term is 9 $\frac{10}{12}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 9 $\frac{10}{12}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9 (odd numbers) and the numerators go 2, 4, 6, 8, 10 (even numbers), all over 12. So there are 5 terms.

1\tfrac{2}{12},\,3\tfrac{4}{12},\,5\tfrac{6}{12},\,7\tfrac{8}{12},\,9\tfrac{10}{12}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 = 25.

1 + 3 + 5 + 7 + 9 = 25

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 12):

\frac{2 + 4 + 6 + 8 + 10}{12}

=

\frac{30}{12}

. Since

\frac{30}{12}

= 2 with remainder 6, this is 2

\frac{6}{12}

.

\dfrac{2 + 4 + 6 + 8 + 10}{12}=\dfrac{30}{12}=2\tfrac{6}{12}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 25 + 2

\frac{6}{12}

= 27

\frac{6}{12}

.

25+2\tfrac{6}{12}=27\tfrac{6}{12}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

27\frac{1}{2}

4 · Reviewdoes it hold up?

There are 5 terms; the whole parts alone sum to 25, and the fraction parts add only 2 $\frac{6}{12}$ more, landing on 27 $\frac{6}{12}$ . The final fraction part $\frac{6}{12}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 27 $\frac{6}{12}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 25.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{30}{12}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 12ths!

Variant 5 medium answer:

39\frac{3}{13}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{13},\ 3\tfrac{4}{13},\ 5\tfrac{6}{13},\ \ldots,\ 11\tfrac{12}{13}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{13}$ , 3 $\frac{4}{13}$ , 5 $\frac{6}{13}$ , ..., up to 11 $\frac{12}{13}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 13.
The last listed term is 11 $\frac{12}{13}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 11 $\frac{12}{13}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12 (even numbers), all over 13. So there are 6 terms.

1\tfrac{2}{13},\,3\tfrac{4}{13},\,5\tfrac{6}{13},\,7\tfrac{8}{13},\,9\tfrac{10}{13},\,11\tfrac{12}{13}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 = 36.

1 + 3 + 5 + 7 + 9 + 11 = 36

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 13):

\frac{2 + 4 + 6 + 8 + 10 + 12}{13}

=

\frac{42}{13}

. Since

\frac{42}{13}

= 3 with remainder 3, this is 3

\frac{3}{13}

.

\dfrac{2 + 4 + 6 + 8 + 10 + 12}{13}=\dfrac{42}{13}=3\tfrac{3}{13}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 36 + 3

\frac{3}{13}

= 39

\frac{3}{13}

.

36+3\tfrac{3}{13}=39\tfrac{3}{13}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

39\frac{3}{13}

4 · Reviewdoes it hold up?

There are 6 terms; the whole parts alone sum to 36, and the fraction parts add only 3 $\frac{3}{13}$ more, landing on 39 $\frac{3}{13}$ . The final fraction part $\frac{3}{13}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 39 $\frac{3}{13}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 36.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{42}{13}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 13ths!

Variant 6 medium answer:

39

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{14},\ 3\tfrac{4}{14},\ 5\tfrac{6}{14},\ \ldots,\ 11\tfrac{12}{14}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{14}$ , 3 $\frac{4}{14}$ , 5 $\frac{6}{14}$ , ..., up to 11 $\frac{12}{14}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 14.
The last listed term is 11 $\frac{12}{14}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 11 $\frac{12}{14}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12 (even numbers), all over 14. So there are 6 terms.

1\tfrac{2}{14},\,3\tfrac{4}{14},\,5\tfrac{6}{14},\,7\tfrac{8}{14},\,9\tfrac{10}{14},\,11\tfrac{12}{14}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 = 36.

1 + 3 + 5 + 7 + 9 + 11 = 36

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 14):

\frac{2 + 4 + 6 + 8 + 10 + 12}{14}

=

\frac{42}{14}

. Since

\frac{42}{14}

= 3 with remainder 0, this is 3.

\dfrac{2 + 4 + 6 + 8 + 10 + 12}{14}=\dfrac{42}{14}=3

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 36 + 3 = 39.

36+3=39

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

39

4 · Reviewdoes it hold up?

There are 6 terms; the whole parts alone sum to 36, and the fraction parts add only 3 more, landing on 39. The final fraction part $\frac{0}{14}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 39.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 36.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{42}{14}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 14ths!

Variant 7 medium answer:

52\frac{11}{15}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{15},\ 3\tfrac{4}{15},\ 5\tfrac{6}{15},\ \ldots,\ 13\tfrac{14}{15}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{15}$ , 3 $\frac{4}{15}$ , 5 $\frac{6}{15}$ , ..., up to 13 $\frac{14}{15}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 15.
The last listed term is 13 $\frac{14}{15}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 13 $\frac{14}{15}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11, 13 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12, 14 (even numbers), all over 15. So there are 7 terms.

1\tfrac{2}{15},\,3\tfrac{4}{15},\,5\tfrac{6}{15},\,7\tfrac{8}{15},\,9\tfrac{10}{15},\,11\tfrac{12}{15},\,13\tfrac{14}{15}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.

1 + 3 + 5 + 7 + 9 + 11 + 13 = 49

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 15):

\frac{2 + 4 + 6 + 8 + 10 + 12 + 14}{15}

=

\frac{56}{15}

. Since

\frac{56}{15}

= 3 with remainder 11, this is 3

\frac{11}{15}

.

\dfrac{2 + 4 + 6 + 8 + 10 + 12 + 14}{15}=\dfrac{56}{15}=3\tfrac{11}{15}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 49 + 3

\frac{11}{15}

= 52

\frac{11}{15}

.

49+3\tfrac{11}{15}=52\tfrac{11}{15}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

52\frac{11}{15}

4 · Reviewdoes it hold up?

There are 7 terms; the whole parts alone sum to 49, and the fraction parts add only 3 $\frac{11}{15}$ more, landing on 52 $\frac{11}{15}$ . The final fraction part $\frac{11}{15}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 52 $\frac{11}{15}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 49.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{56}{15}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 15ths!

Variant 8 hard answer:

68\frac{4}{17}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{17},\ 3\tfrac{4}{17},\ 5\tfrac{6}{17},\ \ldots,\ 15\tfrac{16}{17}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{17}$ , 3 $\frac{4}{17}$ , 5 $\frac{6}{17}$ , ..., up to 15 $\frac{16}{17}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 17.
The last listed term is 15 $\frac{16}{17}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 15 $\frac{16}{17}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11, 13, 15 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12, 14, 16 (even numbers), all over 17. So there are 8 terms.

1\tfrac{2}{17},\,3\tfrac{4}{17},\,5\tfrac{6}{17},\,7\tfrac{8}{17},\,9\tfrac{10}{17},\,11\tfrac{12}{17},\,13\tfrac{14}{17},\,15\tfrac{16}{17}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 17):

\frac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16}{17} = \frac{72}{17}

. Since

\frac{72}{17} = 4

with remainder 4, this is 4

\frac{4}{17}

.

\dfrac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16}{17}=\dfrac{72}{17}=4\tfrac{4}{17}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total:

64 + 4\frac{4}{17} = 68\frac{4}{17}

.

64+4\tfrac{4}{17}=68\tfrac{4}{17}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

68\frac{4}{17}

4 · Reviewdoes it hold up?

There are 8 terms; the whole parts alone sum to 64, and the fraction parts add only 4 $\frac{4}{17}$ more, landing on 68 $\frac{4}{17}$ . The final fraction part $\frac{4}{17}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 68 $\frac{4}{17}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 64.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{72}{17}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 17ths!

Variant 9 hard answer:

85\frac{14}{19}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{19},\ 3\tfrac{4}{19},\ 5\tfrac{6}{19},\ \ldots,\ 17\tfrac{18}{19}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{19}$ , 3 $\frac{4}{19}$ , 5 $\frac{6}{19}$ , ..., up to 17 $\frac{18}{19}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 19.
The last listed term is 17 $\frac{18}{19}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 17 $\frac{18}{19}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11, 13, 15, 17 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12, 14, 16, 18 (even numbers), all over 19. So there are 9 terms.

1\tfrac{2}{19},\,3\tfrac{4}{19},\,5\tfrac{6}{19},\,7\tfrac{8}{19},\,9\tfrac{10}{19},\,11\tfrac{12}{19},\,13\tfrac{14}{19},\,15\tfrac{16}{19},\,17\tfrac{18}{19}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 19):

\frac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18}{19}

=

\frac{90}{19}

. Since

\frac{90}{19}

= 4 with remainder 14, this is 4

\frac{14}{19}

.

\dfrac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18}{19}=\dfrac{90}{19}=4\tfrac{14}{19}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 81 + 4

\frac{14}{19}

= 85

\frac{14}{19}

.

81+4\tfrac{14}{19}=85\tfrac{14}{19}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

85\frac{14}{19}

4 · Reviewdoes it hold up?

There are 9 terms; the whole parts alone sum to 81, and the fraction parts add only 4 $\frac{14}{19}$ more, landing on 85 $\frac{14}{19}$ . The final fraction part $\frac{14}{19}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 85 $\frac{14}{19}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 81.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{90}{19}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 19ths!

Variant 10 hard answer:

105\frac{5}{21}

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\tfrac{2}{21},\ 3\tfrac{4}{21},\ 5\tfrac{6}{21},\ \ldots,\ 19\tfrac{20}{21}$

Show solution

1 · Understandwhat's really being asked

A list of mixed numbers follows a rule: 1 $\frac{2}{21}$ , 3 $\frac{4}{21}$ , 5 $\frac{6}{21}$ , ..., up to 19 $\frac{20}{21}$ . I must figure out the rule, list every term, and add them all up.

Givens

All fraction parts share the denominator 21.
The last listed term is 19 $\frac{20}{21}$ .
The whole parts are odd numbers and the numerators are even numbers.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that fit the rule up to 19 $\frac{20}{21}$ are included.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all terms, then split the sum into a whole-number part and a fraction part -- two easy subproblems.

3 · Execute4 carry out the plan

1Find the rule and list the terms

#5 Look for a Pattern 4.OA.C.5

The whole parts go 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 (even numbers), all over 21. So there are 10 terms.

1\tfrac{2}{21},\,3\tfrac{4}{21},\,5\tfrac{6}{21},\,7\tfrac{8}{21},\,9\tfrac{10}{21},\,11\tfrac{12}{21},\,13\tfrac{14}{21},\,15\tfrac{16}{21},\,17\tfrac{18}{21},\,19\tfrac{20}{21}

Each whole part jumps by 2 and each numerator jumps by 2, making the pattern clear.

2Add the whole-number parts

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100

Summing the odd whole parts is a quick whole-number addition.

3Add the fraction parts

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 21):

\frac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20}{21}

=

\frac{110}{21}

. Since

\frac{110}{21}

= 5 with remainder 5, this is 5

\frac{5}{21}

.

\dfrac{2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20}{21}=\dfrac{110}{21}=5\tfrac{5}{21}

Like denominators let you add numerators, then regroup any improper fraction into a mixed number.

4Combine the two parts

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 100 + 5

\frac{5}{21}

= 105

\frac{5}{21}

.

100+5\tfrac{5}{21}=105\tfrac{5}{21}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer:

105\frac{5}{21}

4 · Reviewdoes it hold up?

There are 10 terms; the whole parts alone sum to 100, and the fraction parts add only 5 $\frac{5}{21}$ more, landing on 105 $\frac{5}{21}$ . The final fraction part $\frac{5}{21}$ is proper, so the answer is sensible.

Another way: Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 105 $\frac{5}{21}$ .

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all the terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts to 100.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping $\frac{110}{21}$ into a mixed number.

💡Takeaway. This only needs Grade 4 pattern-spotting plus same-denominator fraction adding -- split the wholes from the 21ths!

Find patterns in mixed-number sequences and sum · 10 practice problems

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