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

Variant 1 answer: 68 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

Understand

A list of mixed numbers follows a rule: 1 2/17, 3 4/17, 5 6/17, ..., up to 15 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 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 16/17 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 17): (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16)/17 = 72/17. Since 72/17 = 4 with remainder 4, this is 4 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 64 + 4 4/17 = 68 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 4/17

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 68 4/17.

Standards · min 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 72/17 into a mixed number.

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

Variant 2 answer: 27 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

Understand

A list of mixed numbers follows a rule: 1 2/11, 3 4/11, 5 6/11, ..., up to 9 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 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 10/11 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 11): (2 + 4 + 6 + 8 + 10)/11 = 30/11. Since 30/11 = 2 with remainder 8, this is 2 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 25 + 2 8/11 = 27 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 8/11

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 27 8/11.

Standards · min 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 30/11 into a mixed number.

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

Variant 3 answer: 27 6/12

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

Understand

A list of mixed numbers follows a rule: 1 2/12, 3 4/12, 5 6/12, ..., up to 9 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 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 10/12 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 12): (2 + 4 + 6 + 8 + 10)/12 = 30/12. Since 30/12 = 2 with remainder 6, this is 2 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 25 + 2 6/12 = 27 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 6/12

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 27 6/12.

Standards · min 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 30/12 into a mixed number.

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

Variant 4 answer: 85 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

Understand

A list of mixed numbers follows a rule: 1 2/19, 3 4/19, 5 6/19, ..., up to 17 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 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 18/19 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 19): (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18)/19 = 90/19. Since 90/19 = 4 with remainder 14, this is 4 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 81 + 4 14/19 = 85 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 14/19

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 85 14/19.

Standards · min 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 90/19 into a mixed number.

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

Variant 5 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

Understand

A list of mixed numbers follows a rule: 1 2/14, 3 4/14, 5 6/14, ..., up to 11 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 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 12/14 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 14): (2 + 4 + 6 + 8 + 10 + 12)/14 = 42/14. Since 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.

#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

Review

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 0/14 is proper, so the answer is sensible.

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

Standards · min 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 42/14 into a mixed number.

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

Variant 6 answer: 18 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

Understand

A list of mixed numbers follows a rule: 1 2/9, 3 4/9, 5 6/9, ..., up to 7 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 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 8/9 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 9): (2 + 4 + 6 + 8)/9 = 20/9. Since 20/9 = 2 with remainder 2, this is 2 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 16 + 2 2/9 = 18 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 2/9

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 18 2/9.

Standards · min 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 20/9 into a mixed number.

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

Variant 7 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

Understand

A list of mixed numbers follows a rule: 1 2/10, 3 4/10, 5 6/10, ..., up to 7 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 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 8/10 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 10): (2 + 4 + 6 + 8)/10 = 20/10. Since 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.

#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

Review

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 0/10 is proper, so the answer is sensible.

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

Standards · min 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 20/10 into a mixed number.

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

Variant 8 answer: 105 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

Understand

A list of mixed numbers follows a rule: 1 2/21, 3 4/21, 5 6/21, ..., up to 19 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 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 20/21 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 21): (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20)/21 = 110/21. Since 110/21 = 5 with remainder 5, this is 5 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 100 + 5 5/21 = 105 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 5/21

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 105 5/21.

Standards · min 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 110/21 into a mixed number.

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

Variant 9 answer: 39 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

Understand

A list of mixed numbers follows a rule: 1 2/13, 3 4/13, 5 6/13, ..., up to 11 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 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 12/13 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 13): (2 + 4 + 6 + 8 + 10 + 12)/13 = 42/13. Since 42/13 = 3 with remainder 3, this is 3 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 36 + 3 3/13 = 39 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 3/13

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 39 3/13.

Standards · min 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 42/13 into a mixed number.

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

Variant 10 answer: 52 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

Understand

A list of mixed numbers follows a rule: 1 2/15, 3 4/15, 5 6/15, ..., up to 13 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 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 14/15 are included.

Plan

#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.

Execute

#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.

#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.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 15): (2 + 4 + 6 + 8 + 10 + 12 + 14)/15 = 56/15. Since 56/15 = 3 with remainder 11, this is 3 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.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 49 + 3 11/15 = 52 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 11/15

Review

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

Pair the smallest and largest terms (tool 15) and add equal pairs; the running total still comes to 52 11/15.

Standards · min 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 56/15 into a mixed number.

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

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

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