Adding & Subtracting mixed numbers
🎯 In this topic you will
- Add mixed numbers
- Subtract mixed numbers
🧠 Key Words
- cancel
- collecting like terms
- equivalent fractions
- like terms
- numerator
- proper fraction
- simplest form
Show Definitions
- cancel: To simplify a fraction by dividing the numerator and denominator by the same factor.
- collecting like terms: Combining terms in an expression that have the same variable and power.
- equivalent fractions: Fractions that represent the same value even if they look different (e.g., 1/2 = 2/4).
- like terms: Terms in an algebraic expression that have the same variables raised to the same power.
- numerator: The top number of a fraction, showing how many parts are taken.
- proper fraction: A fraction where the numerator is smaller than the denominator.
- simplest form: A fraction that has been reduced so that numerator and denominator have no common factors other than 1.
You already know that you can only add fractions when the denominators are the same.
When the denominators are different, you must write the fractions as equivalent fractions with a common denominator, then add the numerators.
Here is a method for adding mixed numbers, and estimating the answer.
Step ①: Add the whole-number parts. Use this answer as your estimate.
Step ②: Add the fractional parts and cancel this answer to its simplest form.
If this answer is an improper fraction, write it as a mixed number.
Step ③: Add your answers to steps ① and ②.
❓ EXERCISES
1. Copy and complete these additions. Write down an estimate for each of the additions first.
a. $2 \tfrac{4}{9} + 1 \tfrac{4}{9}$
Estimate: $2 + 1 = 3$
Exact: $2 \tfrac{4}{9} + 1 \tfrac{4}{9} = 3 + \tfrac{8}{9} = 3 \tfrac{8}{9}$
b. $7 \tfrac{1}{8} + 3 \tfrac{3}{8}$
Estimate: $7 + 3 = 10$
Exact: $7 \tfrac{1}{8} + 3 \tfrac{3}{8} = 10 + \tfrac{4}{8} = 10 \tfrac{1}{2}$
c. $1 \tfrac{5}{7} + 6 \tfrac{4}{7}$
Estimate: $1 + 6 = 7$
Exact: $1 \tfrac{5}{7} + 6 \tfrac{4}{7} = 7 + \tfrac{9}{7} = 7 + 1 \tfrac{2}{7} = 8 \tfrac{2}{7}$
👀 Show answers
Answers (Q1):
- a. $3 \tfrac{8}{9}$
- b. $10 \tfrac{1}{2}$
- c. $8 \tfrac{2}{7}$
2. The diagram shows the lengths of the three sides of a triangle.
Sides: $3 \tfrac{2}{3}$ m, $5 \tfrac{5}{9}$ m, $7 \tfrac{8}{9}$ m
Work out the perimeter of the triangle. Write your answer as a mixed number, in its simplest form.
👀 Show answer
Answer (Q2):
Convert to improper fractions:
$3 \tfrac{2}{3} = \tfrac{11}{3}$, $5 \tfrac{5}{9} = \tfrac{50}{9}$, $7 \tfrac{8}{9} = \tfrac{71}{9}$
Common denominator = 9
$\tfrac{11}{3} = \tfrac{33}{9}$
Total perimeter = $\tfrac{33}{9} + \tfrac{50}{9} + \tfrac{71}{9} = \tfrac{154}{9}$
= $17 \tfrac{1}{9}$ m
3. Copy and complete these additions. Write down an estimate for each of the additions first.
a. $2 \tfrac{1}{2} + 1 \tfrac{1}{4}$
Estimate: $2 + 1 = 3$
Exact: $2 \tfrac{1}{2} + 1 \tfrac{1}{4} = 3 + \tfrac{3}{4} = 3 \tfrac{3}{4}$
b. $5 \tfrac{1}{3} + 2 \tfrac{1}{6}$
Estimate: $5 + 2 = 7$
Exact: $5 \tfrac{1}{3} + 2 \tfrac{1}{6} = 7 + \tfrac{1}{2} = 7 \tfrac{1}{2}$
c. $1 \tfrac{5}{12} + 3 \tfrac{3}{4}$
Estimate: $1 + 3 = 4$
Exact: $1 \tfrac{5}{12} + 3 \tfrac{3}{4} = 4 + \tfrac{5}{12} + \tfrac{9}{12} = 4 + \tfrac{14}{12} = 4 + 1 \tfrac{2}{12} = 5 \tfrac{1}{6}$
👀 Show answers
Answers (Q3):
- a. $3 \tfrac{3}{4}$
- b. $7 \tfrac{1}{2}$
- c. $5 \tfrac{1}{6}$
4. Andrew uses these two pieces of wood to make a shelf.
Lengths: $2 \tfrac{5}{8}$ m and $1 \tfrac{3}{4}$ m
a. What is the total length of the shelf?
b. Andrew has a wall that is $4 \tfrac{1}{2}$ m long. Will the shelf fit on this wall? Explain your answer.
👀 Show answers
Answer (Q4):
$2 \tfrac{5}{8} + 1 \tfrac{3}{4} = \tfrac{21}{8} + \tfrac{7}{4} = \tfrac{21}{8} + \tfrac{14}{8} = \tfrac{35}{8} = 4 \tfrac{3}{8}$ m
a. Total length of shelf = $4 \tfrac{3}{8}$ m
b. The wall is $4 \tfrac{1}{2}$ m. Since $4 \tfrac{3}{8} < 4 \tfrac{1}{2}$, the shelf will fit on the wall.
5. Hoa drives $12 \tfrac{3}{4}$ km from her home to the doctor’s clinic. She then drives $5 \tfrac{2}{3}$ km from the clinic to her place of work.
Question: What is the total distance that she drives?
🔎 Reasoning Tip
Adding fractions: To add \( \tfrac{3}{4} \) and \( \tfrac{2}{3} \), use the common denominator 12.
👀 Show answer
Answer (Q5):
$12 \tfrac{3}{4} + 5 \tfrac{2}{3} = \tfrac{51}{4} + \tfrac{17}{3}$
LCM of denominators (4, 3) = 12
$\tfrac{51}{4} = \tfrac{153}{12}$ and $\tfrac{17}{3} = \tfrac{68}{12}$
$\tfrac{153}{12} + \tfrac{68}{12} = \tfrac{221}{12} = 18 \tfrac{5}{12}$
Total distance = $18 \tfrac{5}{12}$ km
6. Kia’s homework:
Question: Work out $5 \tfrac{3}{8} + 7 \tfrac{5}{6}$.
1 $5 + 7 = 12$
2 $\tfrac{3}{8} + \tfrac{5}{6} = \tfrac{9}{24} + \tfrac{20}{24} = 1 \tfrac{4}{24} = 1 \tfrac{1}{6}$
3 $12 + 1 \tfrac{1}{6} = 13 \tfrac{1}{6}$
She worked out $5 \tfrac{3}{8} + 7 \tfrac{5}{6}$ but made a mistake.
a. Explain the mistake that Kia has made.
b. Work out the correct answer.
🔎 Reasoning Tip
Finding mistakes: If you cannot see Kia’s mistake, work through the question yourself and then compare your solution with hers.
👀 Show answer
Answer (Q6):
a. Kia’s mistake: She used the wrong denominator. She converted $\tfrac{3}{8}$ to $\tfrac{9}{24}$ correctly but incorrectly converted $\tfrac{5}{6}$ to $\tfrac{20}{24}$ instead of $\tfrac{20}{24} = \tfrac{5}{6}$ (this is fine), but she added them incorrectly as $1 \tfrac{1}{6}$ instead of the true fraction.
b. Correct working:
$5 \tfrac{3}{8} + 7 \tfrac{5}{6} = \tfrac{43}{8} + \tfrac{47}{6}$
LCM of denominators (8, 6) = 24
$\tfrac{43}{8} = \tfrac{129}{24}$ and $\tfrac{47}{6} = \tfrac{188}{24}$
Total = $\tfrac{129}{24} + \tfrac{188}{24} = \tfrac{317}{24}$
= $13 \tfrac{5}{24}$
Correct answer: $13 \tfrac{5}{24}$
7. In this pyramid, you find the mixed number in each block by adding the mixed numbers in the two blocks below it. One addition is shown. Copy and complete the pyramid.
👀 show answer
Bottom row (given): $1\dfrac{1}{4},\; 3\dfrac{2}{3},\; 2\dfrac{5}{6},\; 4\dfrac{4}{9}$.
Second row:
$1\dfrac{1}{4}+3\dfrac{2}{3}=4\dfrac{11}{12}\ (\text{given}),$
$3\dfrac{2}{3}+2\dfrac{5}{6}=6\dfrac{1}{2},$
$2\dfrac{5}{6}+4\dfrac{4}{9}=7\dfrac{5}{18}.$
Third row:
$4\dfrac{11}{12}+6\dfrac{1}{2}=11\dfrac{5}{12},$
$6\dfrac{1}{2}+7\dfrac{5}{18}=13\dfrac{7}{9}.$
Top:
$11\dfrac{5}{12}+13\dfrac{7}{9}=25\dfrac{7}{36}.$
8. Use Leah’s method to simplify these expressions by collecting like terms.
a. $1\dfrac{3}{4}x + 3\dfrac{3}{4}x$
b. $2\dfrac{1}{2}y + 2\dfrac{3}{5}x + 6\dfrac{3}{5}y$
c. $5\dfrac{7}{8}a + 6\dfrac{4}{7}b + 2\dfrac{2}{3}a + 2\dfrac{1}{2}b$
d. $1\dfrac{1}{5}p + 2\dfrac{3}{8}q + \dfrac{2}{3}p + 7\dfrac{4}{5}q$
👀 show answer
a. $1\dfrac{3}{4}x + 3\dfrac{3}{4}x = 5\dfrac{1}{2}x$
b. Collect $y$–terms and $x$–terms:
$2\dfrac{1}{2}y + 6\dfrac{3}{5}y = 9\dfrac{1}{10}y,$ so the expression simplifies to $2\dfrac{3}{5}x + 9\dfrac{1}{10}y$.
c. $a$–terms: $5\dfrac{7}{8}a + 2\dfrac{2}{3}a = 8\dfrac{13}{24}a$.
$b$–terms: $6\dfrac{4}{7}b + 2\dfrac{1}{2}b = 9\dfrac{1}{14}b$.
Result: $8\dfrac{13}{24}a + 9\dfrac{1}{14}b$.
d. $p$–terms: $1\dfrac{1}{5}p + \dfrac{2}{3}p = 1\dfrac{13}{15}p$.
$q$–terms: $2\dfrac{3}{8}q + 7\dfrac{4}{5}q = 10\dfrac{7}{40}q$.
Result: $1\dfrac{13}{15}p + 10\dfrac{7}{40}q$.
🧠 Think like a Mathematician
9. Zara is looking at the question $5\dfrac{2}{3}+7\dfrac{7}{8}$.
- a) Is Zara correct that “without adding any of the fractions, the answer will be between 12 and 14”? Explain.
- b) Choose two mixed numbers (don’t add yet). Copy and complete:
“When I add together my two mixed numbers, the total will be between [ ] and [ ].”Then check by adding them.
- c) For any two mixed numbers, write a general rule for the two whole numbers the total lies between.
- d) How does your rule change for adding three, four or five mixed numbers?
👀 show answer
a) Yes. Each mixed number is between its whole part and the next whole: $5\dfrac{2}{3}\in(5,6)$ and $7\dfrac{7}{8}\in(7,8)$, so their sum is in $(5+7,\;6+8)=(12,14)$. In fact, a tighter bound is between $13$ and $14$ because $\dfrac{2}{3}+\dfrac{7}{8}>1$. Exact sum: $5\dfrac{2}{3}+7\dfrac{7}{8} =12+\dfrac{2}{3}+\dfrac{7}{8} =12+\dfrac{16}{24}+\dfrac{21}{24} =12+\dfrac{37}{24} =13\dfrac{13}{24}\approx13.54$.
b) Method: add the whole parts only to get the lower bound; add “one more” for each mixed number to get the upper bound.
Example choice: $3\dfrac{1}{4}$ and $5\dfrac{2}{5}$ → between $3+5=8$ and $(3+1)+(5+1)=10$. Check: $3\dfrac{1}{4}+5\dfrac{2}{5}=8+\dfrac{1}{4}+\dfrac{2}{5} =8+\dfrac{5}{20}+\dfrac{8}{20}=8\dfrac{13}{20}$, which is indeed between $8$ and $10$.
c)General rule (two mixed numbers): If $a=A+\alpha$ and $b=B+\beta$ with $A,B\in\mathbb{Z}$ and $0<\alpha,\beta<1$, then $$a+b\ \text{is between}\ A+B\ \text{and}\ A+B+2.$$ (Lower bound = sum of whole parts; upper bound = sum of the next whole numbers.)
d) For $n$ mixed numbers, the total lies between “sum of their whole parts” and “that sum + n”. Example for three mixed numbers: between $A_1+A_2+A_3$ and $A_1+A_2+A_3+3$.
❓ EXERCISES
10. $\text{Work out the perimeter of this quadrilateral.}$
🔎 Reasoning Tip
Improper fractions: First, change the improper fractions to mixed numbers.
👀 show answer
$\text{Perimeter} = \dfrac{9}{4} + \dfrac{14}{3} + \dfrac{29}{9} + \dfrac{19}{6}\ \text{m}$
$= \dfrac{81}{36} + \dfrac{168}{36} + \dfrac{116}{36} + \dfrac{114}{36}\ \text{m}$
$= \dfrac{479}{36}\ \text{m} \;=\; 13\dfrac{11}{36}\ \text{m}.$
🍬 Learning Bridge
You’ve just added mixed numbers by estimating with whole parts, then combining fractional parts using a common denominator and simplifying. Next comes subtracting mixed numbers — same toolkit, new twist. You’ll still find a common denominator and simplify, but you may need to regroup (borrow) between the whole number and fraction or switch to improper fractions and back. Keep using quick estimates with the whole parts to check your final answer makes sense.
You already know that you can only subtract fractions when the denominators are the same.
If the denominators are different, you must write the fractions as equivalent fractions with a common denominator, then subtract the numerators.
Here is a method for subtracting mixed numbers.
- 1 Change each mixed number into an improper fraction.
- 2 Subtract the improper fractions and cancel this answer to its simplest form.
- 3 If the answer is an improper fraction, change it back to a mixed number.
❓ EXERCISES
11. Copy and complete these subtractions.
a. $5\dfrac{1}{3} - 2\dfrac{2}{3}$
b. $9\dfrac{1}{6} - 3\dfrac{5}{12}$
c. $5\dfrac{3}{4} - 3\dfrac{5}{6}$
d. $4\dfrac{1}{4} - 1\dfrac{3}{5}$
12. Work out these subtractions. Show all the steps in your working.
a. $2\dfrac{3}{8} - 1\dfrac{5}{8}$
b. $3\dfrac{3}{5} - 1\dfrac{7}{10}$
c. $4\dfrac{2}{3} - 1\dfrac{11}{12}$
d. $5\dfrac{2}{3} - 3\dfrac{1}{4}
👀 show answer
11a. $5\dfrac{1}{3} - 2\dfrac{2}{3} = \dfrac{16}{3} - \dfrac{8}{3} = \dfrac{8}{3} = 2\dfrac{2}{3}$
11b. $9\dfrac{1}{6} - 3\dfrac{5}{12} = \dfrac{55}{6} - \dfrac{41}{12} = \dfrac{110}{12} - \dfrac{41}{12} = \dfrac{69}{12} = \dfrac{23}{4} = 5\dfrac{3}{4}$
11c. $5\dfrac{3}{4} - 3\dfrac{5}{6} = \dfrac{23}{4} - \dfrac{23}{6} = \dfrac{69}{12} - \dfrac{46}{12} = \dfrac{23}{12} = 1\dfrac{11}{12}$
11d. $4\dfrac{1}{4} - 1\dfrac{3}{5} = \dfrac{17}{4} - \dfrac{8}{5} = \dfrac{85}{20} - \dfrac{32}{20} = \dfrac{53}{20} = 2\dfrac{13}{20}$
12a. $2\dfrac{3}{8} - 1\dfrac{5}{8} = \dfrac{19}{8} - \dfrac{13}{8} = \dfrac{6}{8} = \dfrac{3}{4}$
12b. $3\dfrac{3}{5} - 1\dfrac{7}{10} = \dfrac{18}{5} - \dfrac{17}{10} = \dfrac{36}{10} - \dfrac{17}{10} = \dfrac{19}{10} = 1\dfrac{9}{10}$
12c. $4\dfrac{2}{3} - 1\dfrac{11}{12} = \dfrac{14}{3} - \dfrac{23}{12} = \dfrac{56}{12} - \dfrac{23}{12} = \dfrac{33}{12} = 2\dfrac{9}{12} = 2\dfrac{3}{4}$
12d. $5\dfrac{2}{3} - 3\dfrac{1}{4} = \dfrac{17}{3} - \dfrac{13}{4} = \dfrac{68}{12} - \dfrac{39}{12} = \dfrac{29}{12} = 2\dfrac{5}{12}$
🔎 Think like a Mathematician
Work with a partner or in a small group to discuss this question. Look at the different methods Anders and Xavier use to work out: $$9\dfrac{4}{7} - 3\dfrac{6}{7}$$
Anders’ method:
Change $9\dfrac{4}{7}$ into: $8+1+\dfrac{4}{7}=8+\dfrac{7}{7}+\dfrac{4}{7}=8\dfrac{11}{7}$
So: $9\dfrac{4}{7} - 3\dfrac{6}{7} = 8\dfrac{11}{7} - 3\dfrac{6}{7}$
$= (8-3) + \left(\dfrac{11}{7} - \dfrac{6}{7}\right)$
$= 5 + \dfrac{5}{7} = 5\dfrac{5}{7}$
Xavier’s method:
Subtract whole numbers: $9-3=6$
Subtract fractions: $\dfrac{4}{7}-\dfrac{6}{7}=-\dfrac{2}{7}$
So: $6-\dfrac{2}{7}=5\dfrac{5}{7}$
- a) What are the advantages and disadvantages of:
i) Anders’ method
ii) Xavier’s method - b) Which method do you prefer: Anders’ method, Xavier’s method, or the worked example? Explain why.
👀 show answer
a)
- Anders’ method: Advantage – makes use of improper fractions so avoids negative results when subtracting fractions. Disadvantage – slightly longer as it involves rewriting mixed numbers.
- Xavier’s method: Advantage – quicker, directly subtracts whole and fractional parts. Disadvantage – requires handling negative fractions carefully ($\tfrac{4}{7}-\tfrac{6}{7}=-\tfrac{2}{7}$).
b) Preference may vary: - Some learners may prefer Anders’ method because it’s systematic and avoids negatives. - Others may prefer Xavier’s method because it is shorter. - Both give the correct answer of $5\dfrac{5}{7}$.
❓ EXERCISES
14. Work out these subtractions. Show all the steps in your working. Use your preferred method.
a. $3\dfrac{3}{14} - 1\dfrac{4}{7}$
b. $7\dfrac{1}{3} - 2\dfrac{7}{12}$
c. $8\dfrac{2}{3} - 4\dfrac{1}{4}$
d. $6\dfrac{7}{12} - 4\dfrac{17}{18}$
👀 show answer
a. $3\dfrac{3}{14} - 1\dfrac{4}{7} = \dfrac{45}{14} - \dfrac{18}{14} = \dfrac{27}{14} = 1\dfrac{13}{14}$
b. $7\dfrac{1}{3} - 2\dfrac{7}{12} = \dfrac{22}{3} - \dfrac{31}{12} = \dfrac{88}{12} - \dfrac{31}{12} = \dfrac{57}{12} = 4\dfrac{9}{12} = 4\dfrac{3}{4}$
c. $8\dfrac{2}{3} - 4\dfrac{1}{4} = \dfrac{26}{3} - \dfrac{17}{4} = \dfrac{104}{12} - \dfrac{51}{12} = \dfrac{53}{12} = 4\dfrac{5}{12}$
d. $6\dfrac{7}{12} - 4\dfrac{17}{18} = \dfrac{79}{12} - \dfrac{89}{18} = \dfrac{237}{36} - \dfrac{178}{36} = \dfrac{59}{36} = 1\dfrac{23}{36}$
🔎 Think like a Mathematician
Marcus is looking at the question: $9 \tfrac{2}{7} - 3 \tfrac{8}{9}$
Marcus says: “Without subtracting the fractions, I know the answer is going to be between 5 and 7.”
a) Is Marcus correct? Explain your answer.
b) Choose two mixed numbers of your own but don’t subtract them yet. Write between which two whole numbers your total will be. Check that your answer is correct.
c) Think of subtracting any two mixed numbers. Write a rule for working out between which two whole numbers the total will be.
d) How would you change this rule if you were subtracting 3, 4 or 5 mixed numbers?
❓ EXERCISES
16. Shen has two pieces of fabric.
One of the pieces is $1\dfrac{3}{4}\,\text{m}$ long. The other is $2\dfrac{3}{8}\,\text{m}$ long.
a. estimate, then
b. calculate, the difference in length between the two pieces of material.
👀 show answer
a (estimate): $1\dfrac{3}{4}\!\approx\!2$, $2\dfrac{3}{8}\!\approx\!2\dfrac{1}{2}$, so difference $\approx \dfrac{1}{2}\,\text{m}$.
b (calculate): $2\dfrac{3}{8} - 1\dfrac{3}{4} = \dfrac{19}{8} - \dfrac{14}{8} = \dfrac{5}{8}\,\text{m}$.
17. Zalika has a length of wood that is $5\dfrac{1}{4}\,\text{m}$ long.
First, Zalika cuts a piece of wood $1\dfrac{3}{5}\,\text{m}$ long from the length of wood.
Then she cuts a piece of wood $2\dfrac{9}{10}\,\text{m}$ long from the piece of wood she has left.
How long is the piece of wood that Zalika has left over?
👀 show answer
Start: $5\dfrac{1}{4} = \dfrac{105}{20}$.
Subtract $1\dfrac{3}{5} = \dfrac{32}{20}$: $\dfrac{105}{20} - \dfrac{32}{20} = \dfrac{73}{20}$.
Subtract $2\dfrac{9}{10} = \dfrac{58}{20}$: $\dfrac{73}{20} - \dfrac{58}{20} = \dfrac{15}{20} = \dfrac{3}{4}\,\text{m}$.
18. The diagram shows the lengths of the three sides of a triangle.
a. estimate, then
b. calculate, the difference in length between the longest and shortest sides of the triangle.
Write your answer to part b as a mixed number in its simplest form.
👀 show answer
a (estimate): Longest side $\approx 8$, shortest side $\approx 4$, difference $\approx 4$.
b (calculate): Longest $7\dfrac{3}{4} = \dfrac{31}{4}$, shortest $3\dfrac{2}{3} = \dfrac{11}{3}$.
$\dfrac{31}{4} - \dfrac{11}{3} = \dfrac{93 - 44}{12} = \dfrac{49}{12} = 4\dfrac{1}{12}\,\text{m}$.
🔎 Think like a Mathematician
Question: What is the quickest method to use to work out the answer to:
$6 \tfrac{5}{8} - 3 \tfrac{1}{2}$
❓ EXERCISES
20. Sami drives $16\dfrac{5}{8}\,\text{km}$ from his home to work.
Sami drives $11\dfrac{2}{5}\,\text{km}$ from his home to the supermarket.
What is the difference between the distance he drives from his home to work and from his home to the supermarket?
👀 show answer
$16\dfrac{5}{8} = \dfrac{133}{8}, \quad 11\dfrac{2}{5} = \dfrac{57}{5}$
Find common denominator: $\dfrac{133}{8} - \dfrac{57}{5} = \dfrac{665}{40} - \dfrac{456}{40} = \dfrac{209}{40} = 5\dfrac{9}{40}\,\text{km}$.
21. Fina has two bags of lemons.
One bag has a mass of $4\dfrac{7}{10}\,\text{kg}$.
The other bag has a mass of $2\dfrac{4}{15}\,\text{kg}$.
What is the difference in mass between the two bags of lemons?
👀 show answer
$4\dfrac{7}{10} = \dfrac{47}{10}, \quad 2\dfrac{4}{15} = \dfrac{34}{15}$
Find common denominator: $\dfrac{47}{10} - \dfrac{34}{15} = \dfrac{141}{30} - \dfrac{68}{30} = \dfrac{73}{30} = 2\dfrac{13}{30}\,\text{kg}$.
22. This is part of Rio’s homework. He has made a mistake in his solution.
a. Explain the mistake Rio has made.
b. Work out the correct answer.
🔎 Reasoning Tip
Error checking: If you cannot see Rio’s mistake, work through the question yourself and then compare your answer with his.
👀 show answer
a. Rio incorrectly converted $4\dfrac{3}{5}$ to $4\dfrac{6}{10}$ instead of converting the whole fraction properly. He only changed the denominator, not the numerator correctly.
b. Correct calculation:
$4\dfrac{3}{5} = \dfrac{23}{5} = \dfrac{46}{10}$.
$\dfrac{46}{10} - \dfrac{9}{10} = \dfrac{37}{10} = 3\dfrac{7}{10}$.
23. In this pyramid, you find the mixed number in each block by adding the mixed numbers in the two blocks below it.
Complete the pyramid.
👀 show answer
Bottom row: $1\dfrac{5}{9},\; ?,\; ?$
Second row: $1\dfrac{5}{9}+? = 2\dfrac{4}{5} \;\;\Rightarrow ? = 2\dfrac{4}{5} - 1\dfrac{5}{9}$
$2\dfrac{4}{5} = \dfrac{29}{15},\;\; 1\dfrac{5}{9} = \dfrac{14}{9}$
Common denominator $=45$: $\dfrac{87}{45} - \dfrac{70}{45} = \dfrac{17}{45}$
So the missing block is $\dfrac{17}{45}$.
Then second missing block: $? + 2\dfrac{4}{5} = ?$ (leading to $8\dfrac{2}{3}$ in the row above)
We know $8\dfrac{2}{3} - 2\dfrac{4}{5} = 5\dfrac{13}{15}$, so the next missing block is $5\dfrac{13}{15}$.
Row check:
$1\dfrac{5}{9} + \dfrac{17}{45} = 2\dfrac{4}{5}$ ✔️
$\dfrac{17}{45} + 5\dfrac{13}{15} = 6\dfrac{2}{9}$ ✔️
$2\dfrac{4}{5} + 6\dfrac{2}{9} = 8\dfrac{2}{3}$ ✔️
$8\dfrac{2}{3} + 4\dfrac{1}{12} = 12\dfrac{3}{4}$ ✔️
Final completed pyramid:
- Top: $12\dfrac{3}{4}$
- Second row: $8\dfrac{2}{3},\; 4\dfrac{1}{12}$
- Third row: $2\dfrac{4}{5},\; 6\dfrac{2}{9}$
- Bottom row: $1\dfrac{5}{9},\; \dfrac{17}{45},\; 5\dfrac{13}{15}$
24. The perimeter of this quadrilateral is $35\dfrac{13}{36}\,\text{m}$.
The sides are $5\dfrac{1}{9}\,\text{m}$, $8\dfrac{2}{3}\,\text{m}$, $9\dfrac{5}{6}\,\text{m}$, and one missing side.
Work out the length of the missing side.
🔎 Reasoning Tip
Perimeter: The perimeter of a shape is the distance around the edge of the shape.
👀 show answer
Convert each known side to an improper fraction:
- $5\dfrac{1}{9} = \dfrac{46}{9}$
- $8\dfrac{2}{3} = \dfrac{26}{3}$
- $9\dfrac{5}{6} = \dfrac{59}{6}$
- Perimeter $= 35\dfrac{13}{36} = \dfrac{1273}{36}$
Find common denominator $36$:
$\dfrac{46}{9} = \dfrac{184}{36},\;\; \dfrac{26}{3} = \dfrac{312}{36},\;\; \dfrac{59}{6} = \dfrac{354}{36}$
Sum of known sides $= \dfrac{184}{36} + \dfrac{312}{36} + \dfrac{354}{36} = \dfrac{850}{36}$
Missing side $= \dfrac{1273}{36} - \dfrac{850}{36} = \dfrac{423}{36} = \dfrac{141}{12} = 11\dfrac{3}{4}\,\text{m}$
Final Answer: The missing side is $11\dfrac{3}{4}\,\text{m}$.
⚠️ Be careful! Adding & subtracting mixed numbers
- Find a common denominator first — never add or subtract denominators.
Wrong: $\tfrac{1}{4}+\tfrac{5}{6}\neq\tfrac{6}{10}$ Correct: $\tfrac{1}{4}+\tfrac{5}{6}=\tfrac{3}{12}+\tfrac{10}{12}=\tfrac{13}{12}$. - Carry from the fractional sum if it’s $\ge 1$.
$2\tfrac{3}{8}+1\tfrac{5}{6}=3+\tfrac{9}{24}+\tfrac{20}{24}=3+\tfrac{29}{24}=4\tfrac{5}{24}$. - For subtraction, regroup (borrow) or use improper fractions when the top fraction is smaller.
$4\tfrac{1}{6}-2\tfrac{5}{6}=\tfrac{25}{6}-\tfrac{17}{6}=\tfrac{8}{6}=\tfrac{4}{3}=1\tfrac{1}{3}$ (don’t leave a negative fractional part). - Simplify at the end: reduce fractions and mixed numbers to simplest form, e.g. $\tfrac{14}{12}=\tfrac{7}{6}=1\tfrac{1}{6}$.
- Estimate with whole parts to sense‑check your result before/after calculation.
- Collecting like terms: only combine the same variable, and add their (mixed‑number) coefficients correctly.
$1\tfrac{3}{4}x+3\tfrac{3}{4}x=5\tfrac{1}{2}x$ (don’t mix $x$ with $y$).