P6 Algebra Problem Sums ALB10

ALB 10

Comparison Model + Substitution  |  Two-Part Questions

ALB 10 combines everything from ALB 1–9 into two-part questions:
Part (a) — Express the total in terms of the unknown letter.
Part (b) — You are given the actual total. Form an equation, then solve for the unknown by working backwards.
✏️  Guided Example
Jeremy bought 8 identical chairs and 5 identical tables. Each chair cost $m and each table cost $145.
  1. Express the total cost of the chairs and tables in terms of m.
  2. The total cost of the chairs and tables was $1357. Form an equation in terms of m. Then, find the value of m.
STEP 1
Understanding the Problem

What was the total cost of the chairs?

▶ 8 × $m = $8m

What was the total cost of the tables?

▶ 5 × $145 = $725

What am I trying to find for part (a)?

▶ The total cost of chairs and tables in terms of m.

What am I trying to find for part (b)?

▶ The value of m.

STEP 2
Represent the Problem

▶ Draw a comparison model.

8 chairs: $m $m $m $m $m $m $m $m 5 tables: $145 $145 $145 $145 $145 = $8m = $725
STEP 3
Devising a Plan & Carrying Out
Part (a)
8 × $m + 5 × $145
= $8m + $725
= $(8m + 725)
Total cost = $(8m + 725)
Part (b)
8m + 725 = 1357    ← form the equation
8m = 1357 − 725
8m = 632
m = 632 ÷ 8
m = 79
The value of m is 79.
STEP 4
Looking Back
Cost of chair = $79
Total = 8 × $79 + 5 × $145
       = $632 + $725
       = $1357 ✓
💡 Key Idea: Part (a) — write the expression using the letter (no solving needed).
Part (b) — set the expression equal to the given total, then work backwards (subtract the known part, then divide) to find the value of the letter.
ALB 10

Practice – Comparison Model + Substitution (Two-Part)

Score 0 / 8
📝 Each question has two parts: (a) express in terms of the letter  |  (b) form an equation and find the value of the letter. Lock in all answers before revealing the solution.
1

Jack has 15 small packets of chocolates and 25 big packets of lollipops. Each small packet contains 14 chocolates and each big packet contains n lollipops.

  1. How many chocolates and lollipops does Jack have altogether? Give your answer in terms of n.
  2. Jack has a total of 760 chocolates and lollipops. Form an equation in terms of n. Then, find the value of n.
(a)  1 mark
Total in terms of n:
(b)  1 mark
Form an equation and find the value of n:
💡 Set your expression = 760, then solve for n.
✅ Worked Answer
760 15×14=210 25 × n = 25n chocolates lollipops 760 − 210 = 550 = 25n n = 550 ÷ 25 = 22
(a) 15 × 14 + 25 × n = 210 + 25n = 25n + 210
(b) 25n + 210 = 760
     25n = 760 − 210 = 550
     n = 550 ÷ 25 = 22
(a) Total = 25n + 210    (b) n = 22
2

Linda had 22 small boxes of purple hair pins and 35 big boxes of pink hair pins. Each small box contained 18 purple hair pins and each big box contained q pink hair pins. She then gave 333 hair pins to her sister.

  1. How many hair pins did Linda have left? Give your answer in terms of q.
  2. Linda had 1043 hair pins left. Form an equation in terms of q. Then, find the value of q.
(a)  1 mark
Hair pins left in terms of q:
💡 Total first = 22×18 + 35q. Then subtract 333.
(b)  1 mark
Form an equation and find the value of q:
✅ Worked Answer
Total: 22×18=396 35q After: 333 given left = 35q + 63 35q + 63 = 1043 → 35q = 980 → q = 28
(a) Purple = 22 × 18 = 396
     Total = 396 + 35q
     After giving: 396 + 35q − 333 = 35q + 63
(b) 35q + 63 = 1043
     35q = 980 → q = 28
(a) Left = 35q + 63    (b) q = 28
3

Gillian baked 12 small trays of vanilla muffins and 22 big trays of chocolate muffins. Each small tray contained a dozen (12) vanilla muffins and each big tray contained m chocolate muffins.

  1. How many vanilla and chocolate muffins did Gillian bake altogether? Give your answer in terms of m.
  2. Gillian then sold 134 muffins and had 428 muffins left. Form an equation in terms of m. Then, find the value of m.
(a)  1 mark
Total muffins in terms of m:
(b)  1 mark
Form an equation and find the value of m:
💡 Total baked = 428 + 134 = 562. Set 22m + 144 = 562.
✅ Worked Answer
Total = 22m + 144 12×12=144 22m After: 134 sold left = 428 22m + 144 = 562 → 22m = 418 → m = 19
(a) Vanilla = 12 × 12 = 144
     Total = 144 + 22m = 22m + 144
(b) Total baked = 428 + 134 = 562
     22m + 144 = 562 → 22m = 418 → m = 19
(a) Total = 22m + 144    (b) m = 19
4

The average height of a group of 14 pupils was h cm. Ben who was 136 cm tall joined the group.

  1. What was the total height of the new group of pupils? Give your answer in terms of h.
  2. The average height of the new group of pupils was 134.6 cm. Form an equation in terms of h. Then, find the value of h.
(a)  1 mark
Total height of new group in terms of h:
(b)  1 mark
Form an equation and find the value of h:
💡 New group = 15 pupils. Total = 134.6 × 15 = 2019. Set 14h + 136 = 2019.
✅ Worked Answer
Total = 14h + 136 = 2019 14 × h = 14h 136 14 original pupils Ben 14h + 136 = 2019 14h = 1883 → h = 134.5
(a) 14 × h + 136 = 14h + 136
(b) New group = 15 pupils
     Total = 134.6 × 15 = 2019
     14h + 136 = 2019
     14h = 1883 → h = 134.5
(a) Total = 14h + 136    (b) h = 134.5

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