Algebra and equations · Ages 9–10
Two unknowns worksheet
Part of: Patterns and rules
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Two unknowns worksheet
Algebra and equations · Ages 9–10
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.△ + ○ = 19. When △ = 8, ○ = ; when △ = 9, ○ = .
Fill in each blank
Answer:
2.△ + ○ = 14. When △ = 12, ○ =
Answer:
3.△ + ○ = 20. When △ = 7, ○ =
Answer:
4.△ + ○ = 16. When △ = 6, ○ = ; when △ = 7, ○ = .
Fill in each blank
Answer:
5.△ × ○ = 15, where △ and ○ are whole numbers greater than 0 (△ = 1, ○ = 15 and △ = 15, ○ = 1 count as two pairs). How many pairs are there?
Answer:
6.△ + ○ = 8, where △ and ○ are whole numbers from 1 to 9 (△ = 1, ○ = 2 and △ = 2, ○ = 1 count as two pairs). How many pairs are there?
Answer:
7.△ + ○ = 11, where △ and ○ are whole numbers from 1 to 9 (△ = 1, ○ = 2 and △ = 2, ○ = 1 count as two pairs). How many pairs are there?
Answer:
8.△ × ○ = 8, where △ and ○ are whole numbers greater than 0 (△ = 1, ○ = 8 and △ = 8, ○ = 1 count as two pairs). How many pairs are there?
Answer:
9.△ + ○ = 33 and △ − ○ = 1. △ = and ○ = .
Fill in each blank
Answer:
10.△ + ○ = 13 and △ − ○ = 3. △ = and ○ = .
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
11.Which pair fits △ − ○ = 9?
- A. △ = 8, ○ = 17
- B. △ = 17, ○ = 8
- C. △ = 5, ○ = 4
- D. △ = 18, ○ = 8
Choose the right answer
Answer:
12.Which pair fits △ − ○ = 8?
- A. △ = 11, ○ = 2
- B. △ = 10, ○ = 2
- C. △ = 7, ○ = 1
- D. △ = 2, ○ = 10
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A hall must set out all 20 chairs in rows, the same number in each row, with no more rows than chairs in a row. How many different ways are there?
Working:
Answer:
14.Tsz-ching has to pay exactly $13 with $2 and $5 coins (using just one kind is allowed). How many different ways are there?
Working:
Answer:
15.May and Tsz-ching have 58 shells between them, and May has 8 more than Tsz-ching. May has and Tsz-ching has .
Fill in each blank
Working:
Answer:
16.A box of 24 chocolates is shared out between a red bag and a blue bag, with at least 2 in each. How many ways are there?
Working:
Answer:
17.Ka-yan has to pay exactly $14 with $2 and $5 coins (using just one kind is allowed). How many different ways are there?
Working:
Answer:
18.A hall must set out all 36 chairs in rows, the same number in each row, with no more rows than chairs in a row. How many different ways are there?
Working:
Answer:
Show the answers and working
Two unknowns worksheet: Answers and working
Part A: Working out and filling in
1. 11, 10
△ and ○ make 19, so ○ is 19 minus △. Start with △ = 8.
○ is 19 minus △: 19 − 8 = 11, 19 − 9 = 10. A different △ gives a different ○.
2. 2
△ and ○ make 14, so ○ is 14 minus △.
○ is 14 minus △: 14 − 12 = 2, so ○ = 2.
3. 13
△ and ○ make 20, so ○ is 20 minus △.
○ is 20 minus △: 20 − 7 = 13, so ○ = 13.
4. 10, 9
△ and ○ make 16, so ○ is 16 minus △. Start with △ = 6.
○ is 16 minus △: 16 − 6 = 10, 16 − 7 = 9. A different △ gives a different ○.
5. 4
Try △ = 1, 2, 3, … in turn and see which divide 15 exactly.
Try △ = 1, 2, 3, …: only the ones that divide 15 exactly have an ○, so 4 pairs.
6. 7
List them in order: what is the smallest △ can be, and the largest? Remember ○ must be from 1 to 9 too.
List them in order: △ is at least 1 and at most 7, each with one ○, so 7 pairs.
7. 8
List them in order: what is the smallest △ can be, and the largest? Remember ○ must be from 1 to 9 too.
List them in order: △ is at least 2 and at most 9, each with one ○, so 8 pairs.
8. 4
Try △ = 1, 2, 3, … in turn and see which divide 8 exactly.
Try △ = 1, 2, 3, …: only the ones that divide 8 exactly have an ○, so 4 pairs.
9. 17, 16
List the pairs that add up to 33, then look for the pair that is 1 apart.
List the pairs adding to 33 and find the one 1 apart: △ = 17, ○ = 16. Check: 17 + 16 = 33, 17 − 16 = 1.
10. 8, 5
List the pairs that add up to 13, then look for the pair that is 3 apart.
List the pairs adding to 13 and find the one 3 apart: △ = 8, ○ = 5. Check: 8 + 5 = 13, 8 − 5 = 3.
Part B: Choose, or true or false
11. B. △ = 17, ○ = 8
△ − ○ = 9 means △ is 9 more than ○. Work out △ − ○ for each option.
17 − 8 = 9: only this pair fits.
12. B. △ = 10, ○ = 2
△ − ○ = 8 means △ is 8 more than ○. Work out △ − ○ for each option.
10 − 2 = 8: only this pair fits.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 3 ways
Rows × chairs in a row = 20. Try 1 row, 2 rows, …, and stop once the rows would outnumber the chairs in a row.
Rows × chairs a row = 20. Try 1, 2, 3, … rows: a number that divides 20 and isn't more than the chairs a row works. 3 in all.
14. 1 way
List by the number of $5 coins, starting from 0. What is left must be even to pay in $2 coins.
List in order: 0, 1, 2, … $5 coins, and each time see whether the rest can be paid in $2 coins: 1 ways work.
15. 33, 25
If May didn't have the extra 8, they would have the same. Take 8 from 58 first.
Guess and adjust, or subtract the extra 8 so they match: May has 33 and Tsz-ching 25. Check: 33 + 25 = 58, 33 − 25 = 8.
16. 21 ways
The red bag holds at least 2. What is the most it can hold, keeping at least 2 for the blue bag?
The red bag can hold 2, …, 22, and each has exactly one blue amount to match: 21 ways.
17. 2 ways
List by the number of $5 coins, starting from 0. What is left must be even to pay in $2 coins.
List in order: 0, 1, 2, … $5 coins, and each time see whether the rest can be paid in $2 coins: 2 ways work.
18. 5 ways
Rows × chairs in a row = 36. Try 1 row, 2 rows, …, and stop once the rows would outnumber the chairs in a row.
Rows × chairs a row = 36. Try 1, 2, 3, … rows: a number that divides 36 and isn't more than the chairs a row works. 5 in all.
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