Mastery Guide: The Unitary Method, Direct Proportion, Rates & Financial Applications
Instructional Blueprint: Unit 6 Unitary Method
- Understand the meaning of "Unitary" (finding the value of 1 unit first).
- Calculate the value of many items when the value of 1 item is given (Multiplication).
- Calculate the value of 1 item when the value of many items is given (Division).
- Master the full two-step Unitary Method: Given Many → Find 1 → Find Required Many.
- Solve multi-step real-world problems involving shopping, weights, distances, bus capacities, and costs.
- 00–15m: Hook & Intuition: The Vegetable Vendor Story & "Unit = 1".
- 15–30m: Type 1 (Multiply: $1 \to \text{Many}$) & Type 2 (Divide: $\text{Many} \to 1$).
- 30–45m: The 2-Step Master Strategy ($\text{Many} \to 1 \to \text{Target}$).
- 45–60m: Exercise 1 & Exercise 2 Step-by-Step Problem Solving.
- The Stepping Stone Metaphor: "You cannot jump across the river in one leap; step on the stone of $1$ unit first!"
- Memory Rule: "To go to ONE → DIVIDE. To go to MANY → MULTIPLY!"
🛒 Real-Life Challenge: The Vegetable Market Story
Imagine you go to the market with your mother. The vegetable vendor says: "5 kilograms of fresh vegetables cost Rs. 400."
Your mother wants to buy 13 kilograms. How much should she pay?
The Child's Strategy: You cannot jump directly from $5\text{ kg}$ to $13\text{ kg}$ easily. But look how simple it is if you find $1\text{ kg}$ first:
• Step 1 (Find 1 kg): $\text{Price of } 1\text{ kg} = \text{Rs. } 400 \div 5 = \mathbf{\text{Rs. } 80}$
• Step 2 (Find 13 kg): $\text{Price of } 13\text{ kg} = \text{Rs. } 80 \times 13 = \mathbf{\text{Rs. } 1,040}$
This clever method is called the Unitary Method!
1. Type 1: Calculating the Value of Many Objects from One Object
When you know the price or weight of one ($1$) item, finding the total for many items is super easy: simply MULTIPLY ($\times$)!
For a science experiment, students were divided into $8\text{ groups}$ and each group was given $1\text{ magnifying glass}$. If the price of $1\text{ magnifying glass}$ is $\text{Rs. } 245$, find the price of $8\text{ glasses}$.
$$\text{Price of } 1\text{ glass} = \text{Rs. } 245$$
$$\text{Price of } 8\text{ glasses} = 245 \times 8 = \mathbf{\text{Rs. } 1,960}$$
If the price of a pencil is $\text{Rs. } 10$, what will be the price of $15\text{ pencils}$?
$$\text{Price of } 1\text{ pencil} = \text{Rs. } 10$$
$$\text{Price of } 15\text{ pencils} = 10 \times 15 = \mathbf{\text{Rs. } 150}$$
2. Type 2: Calculating the Value of One Object from Many Objects
When you know the total cost or quantity of many items together, finding the cost of just one ($1$) single item requires sharing or distributing equally: simply DIVIDE ($\div$)!
The price of $20\text{ chairs}$ is $\text{Rs. } 7,240$. How can we find the price of $1\text{ such chair}$?
$$\text{Price of } 1\text{ chair} = 7,240 \div 20 = \mathbf{\text{Rs. } 362}$$
In a mango orchard, there are $576\text{ trees}$ in $32\text{ identical rows}$. How many trees will there be in $1\text{ row}$?
$$\text{Number of trees in } 1\text{ row} = 576 \div 32 = \mathbf{18\text{ trees}}$$
🌟 Summary of the Fundamental Unitary Rules:
- Rule 1: When the value of one item is known, the value of many items of the same kind can be found by MULTIPLICATION.
- Rule 2: When the value of many items is known, then the price of one item can be found by DIVISION.
3. Type 3: The Complete Two-Step Unitary Method
In most real-life exam questions, you are given the value of some items ($A$) and asked to find the value of different items ($B$). You use the two-step bridge:
Father bought $12\text{ tube lights}$ for our home which cost $\text{Rs. } 6,900$ altogether. What will be the price of $7\text{ such tube lights}$?
• Step 1: Price of $1\text{ tube light} = 6,900 \div 12 = \text{Rs. } 575$
• Step 2: Price of $7\text{ tube lights} = 575 \times 7 = \mathbf{\text{Rs. } 4,025}$
There are $2,940\text{ pages}$ in $30\text{ notebooks}$. How many pages will be there in $14\text{ such notebooks}$?
• Step 1: Pages in $1\text{ notebook} = 2,940 \div 30 = 98\text{ pages}$
• Step 2: Pages in $14\text{ notebooks} = 98 \times 14 = \mathbf{1,372\text{ pages}}$
📝 100% Solved Textbook Exercises • Grade 5 Mathematics Chapter 6
Given:
Price of $1\text{ geometry box} = \text{Rs. } 76$
Number of geometry boxes $= 26$
Working:
$$\text{Price of } 26\text{ boxes} = 76 \times 26$$
$$\begin{array}{r@{\quad}l}
& 76 \\
\times & 26 \\
\hline
& 456 \quad (76 \times 6) \\
+ & 1520 \quad (76 \times 20) \\
\hline
& \mathbf{1,976}
\end{array}$$
Answer: The price of $26\text{ geometry boxes}$ is $\text{Rs. } 1,976$.
Given:
Price of $1\text{ ice cream} = \text{Rs. } 55$
Number of ice creams $= 17$
Working:
$$\text{Price of } 17\text{ ice creams} = 55 \times 17$$
$$\begin{array}{r@{\quad}l}
& 55 \\
\times & 17 \\
\hline
& 385 \quad (55 \times 7) \\
+ & 550 \quad (55 \times 10) \\
\hline
& \mathbf{935}
\end{array}$$
Answer: The price of $17\text{ ice creams}$ is $\text{Rs. } 935$.
Given:
Price of $1\text{ pack} = \text{Rs. } 37$
Number of packs $= 71$
Working:
$$\text{Total price} = 37 \times 71$$
$$\begin{array}{r@{\quad}l}
& 37 \\
\times & 71 \\
\hline
& 37 \quad (37 \times 1) \\
+ & 2590 \quad (37 \times 70) \\
\hline
& \mathbf{2,627}
\end{array}$$
Answer: The price of $71\text{ popcorn packs}$ is $\text{Rs. } 2,627$.
Given:
Weight of $1\text{ bag} = 21\text{ kg}$
Number of bags $= 15$
Working:
$$\text{Total weight} = 21 \times 15$$
$$\begin{array}{r@{\quad}l}
& 21 \\
\times & 15 \\
\hline
& 105 \quad (21 \times 5) \\
+ & 210 \quad (21 \times 10) \\
\hline
& \mathbf{315}
\end{array}$$
Answer: The total weight of $15\text{ bags}$ is $315\text{ kilograms}$.
Given:
Capacity of $15\text{ buses} = 555\text{ passengers}$
Number of buses $= 15$
Working:
$$\text{Capacity of } 1\text{ bus} = 555 \div 15$$
$$15 \times 30 = 450,\quad 555 - 450 = 105,\quad 15 \times 7 = 105 \implies 30 + 7 = \mathbf{37}$$
Answer: The capacity of $1\text{ bus}$ is $37\text{ passengers}$.
Given:
Price of $26\text{ kg sugar} = \text{Rs. } 1,170$
Quantity $= 26\text{ kg}$
Working:
$$\text{Price of } 1\text{ kg sugar} = 1,170 \div 26$$
$$26 \times 40 = 1,040,\quad 1,170 - 1,040 = 130,\quad 26 \times 5 = 130 \implies 40 + 5 = \mathbf{45}$$
Answer: The price of $1\text{ kilogram of sugar}$ is $\text{Rs. } 45$.
Given:
Rent for $8\text{ months} = \text{Rs. } 145,680$
Number of months $= 8$
Working:
$$\text{Rent for } 1\text{ month} = 145,680 \div 8$$
$$145,680 \div 8 = \mathbf{18,210}$$
Answer: The monthly rent of the house is $\text{Rs. } 18,210$.
Given:
Cost of $35\text{ registers} = \text{Rs. } 5,075$
Number of registers $= 35$
Working:
$$\text{Cost of } 1\text{ register} = 5,075 \div 35$$
$$35 \times 100 = 3,500,\quad 5,075 - 3,500 = 1,575,\quad 35 \times 40 = 1,400,\quad 175 \div 35 = 5 \implies 100 + 40 + 5 = \mathbf{145}$$
Answer: The cost of $1\text{ register}$ is $\text{Rs. } 145$.
Step 1 (Find markers in $1\text{ box}$):
$$\text{Markers in } 1\text{ box} = 4,900 \div 50 = \mathbf{98\text{ markers}}$$
Step 2 (Find markers in $75\text{ boxes}$):
$$\text{Markers in } 75\text{ boxes} = 98 \times 75$$
$$\begin{array}{r@{\quad}l}
& 98 \\
\times & 75 \\
\hline
& 490 \quad (98 \times 5) \\
+ & 6860 \quad (98 \times 70) \\
\hline
& \mathbf{7,350}
\end{array}$$
Answer: There will be $7,350\text{ markers}$ in $75\text{ boxes}$.
Step 1 (Find price of $1\text{ mobile phone}$):
$$\text{Price of } 1\text{ phone} = 348,290 \div 10 = \mathbf{\text{Rs. } 34,829}$$
Step 2 (Find price of $29\text{ mobile phones}$):
$$\text{Price of } 29\text{ phones} = 34,829 \times 29$$
$$\begin{array}{r@{\quad}l}
& 34,829 \\
\times & 29 \\
\hline
& 313,461 \quad (34,829 \times 9) \\
+ & 696,580 \quad (34,829 \times 20) \\
\hline
& \mathbf{1,010,041}
\end{array}$$
Answer: The price of $29\text{ mobile phones}$ is $\text{Rs. } 1,010,041$.
Step 1 (Find weight of $1\text{ book}$):
$$\text{Weight of } 1\text{ book} = 88 \div 40 = \frac{88}{40} = \frac{22}{10} = \mathbf{2.2\text{ kg}}\quad (= 2\text{ kg } 200\text{ g})$$
Step 2 (Find weight of $6\text{ books}$):
$$\text{Weight of } 6\text{ books} = 2.2 \times 6 = \mathbf{13.2\text{ kg}}\quad (= 13\text{ kg } 200\text{ g})$$
Answer: The weight of $6\text{ books}$ is $13.2\text{ kg}$ (or $13\text{ kg } 200\text{ g}$).
Step 1 (Find distance covered in $1\text{ hour}$ / Speed):
$$\text{Distance in } 1\text{ hour} = 6,136 \div 52$$
$$52 \times 100 = 5,200,\quad 6,136 - 5,200 = 936,\quad 52 \times 18 = 936 \implies \mathbf{118\text{ km/h}}$$
Step 2 (Find distance covered in $45\text{ hours}$):
$$\text{Distance in } 45\text{ hours} = 118 \times 45$$
$$\begin{array}{r@{\quad}l}
& 118 \\
\times & 45 \\
\hline
& 590 \quad (118 \times 5) \\
+ & 4720 \quad (118 \times 40) \\
\hline
& \mathbf{5,310}
\end{array}$$
Answer: The train will cover $5,310\text{ kilometres}$ in $45\text{ hours}$.
Step 1 (Find cost of $1\text{ computer}$):
$$\text{Cost of } 1\text{ computer} = 838,155 \div 3 = \mathbf{\text{Rs. } 279,385}$$
Step 2 (Find cost of $52\text{ computers}$):
$$\text{Cost of } 52\text{ computers} = 279,385 \times 52$$
$$\begin{array}{r@{\quad}l}
& 279,385 \\
\times & 52 \\
\hline
& 558,770 \quad (279,385 \times 2) \\
+ & 13,969,250 \quad (279,385 \times 50) \\
\hline
& \mathbf{14,528,020}
\end{array}$$
Answer: The cost of $52\text{ computers}$ is $\text{Rs. } 14,528,020$.
Step 1 (Find fare for $1\text{ kilometre}$):
$$\text{Fare for } 1\text{ km} = 1,190 \div 34 = \mathbf{\text{Rs. } 35}$$
Step 2 (Find fare for $48\text{ kilometres}$):
$$\text{Fare for } 48\text{ km} = 35 \times 48 = \mathbf{\text{Rs. } 1,680}$$
Answer: The fare for $48\text{ kilometres}$ is $\text{Rs. } 1,680$.
Step 1 (Find price of $1\text{ pair of shoes}$):
$$\text{Price of } 1\text{ pair} = 23,750 \div 19 = \mathbf{\text{Rs. } 1,250}$$
Step 2 (Find price of $65\text{ pairs}$):
$$\text{Price of } 65\text{ pairs} = 1,250 \times 65 = \mathbf{\text{Rs. } 81,250}$$
Answer: The price of $65\text{ pairs of shoes}$ is $\text{Rs. } 81,250$.
a) The price of a book is $\text{Rs. } 250$. The price of $5\text{ books}$ will be $\text{Rs. }$ ______.
(i) $50$ (ii) $1,000$ (iii) $1,250$ ✓ (iv) $2,500$
Explanation: $\text{Price} = 250 \times 5 = \mathbf{\text{Rs. } 1,250}$.
b) The price of $11\text{ carpets}$ is $\text{Rs. } 35,805$. The price of one carpet is ______.
(i) $3,855$ (ii) $3,055$ (iii) $2,355$ (iv) $3,255$ ✓
Explanation: $\text{Price of } 1\text{ carpet} = 35,805 \div 11 = \mathbf{\text{Rs. } 3,255}$.
c) The price of a book is $\text{Rs. } 555$. The price of $10\text{ books}$ will be $\text{Rs. }$ ______.
(i) $2,550$ (ii) $5,050$ (iii) $3,250$ (iv) $5,550$ ✓
Explanation: $\text{Price} = 555 \times 10 = \mathbf{\text{Rs. } 5,550}$.
d) The price of $6\text{ oranges}$ is $\text{Rs. } 48$. The price of $72\text{ oranges}$ will be $\text{Rs. }$ ______.
(i) $567$ (ii) $96$ (iii) $112$ (iv) $576$ ✓
Explanation: $1\text{ orange} = 48 \div 6 = \text{Rs. } 8 \implies 72\text{ oranges} = 8 \times 72 = \mathbf{\text{Rs. } 576}$.
e) The price of $3\text{ chairs}$ is $\text{Rs. } 645$. The price of $16\text{ chairs}$ will be $\text{Rs. }$ ______.
(i) $3,040$ (ii) $3,004$ (iii) $3,440$ ✓ (iv) $3,404$
Explanation: $1\text{ chair} = 645 \div 3 = \text{Rs. } 215 \implies 16\text{ chairs} = 215 \times 16 = \mathbf{\text{Rs. } 3,440}$.
Given:
Cost of $1\text{ storybook} = \text{Rs. } 440$
Number of storybooks $= 15$
Working:
$$\text{Price of } 15\text{ storybooks} = 440 \times 15 = \mathbf{\text{Rs. } 6,600}$$
Answer: The price of $15\text{ storybooks}$ is $\text{Rs. } 6,600$.
Given:
Price of $16\text{ bags} = \text{Rs. } 24,000$
Number of bags $= 16$
Working:
$$\text{Price of } 1\text{ bag} = 24,000 \div 16 = \mathbf{\text{Rs. } 1,500}$$
Answer: The price of $1\text{ school bag}$ is $\text{Rs. } 1,500$.
a) How many pearls will be used to make $25\text{ necklaces}$?
$$\text{Pearls} = 425 \times 25 = \mathbf{10,625\text{ pearls}}$$
b) How many pearls will be used to make $100\text{ necklaces}$?
$$\text{Pearls} = 425 \times 100 = \mathbf{42,500\text{ pearls}}$$
First, find the price of $1\text{ washing machine}$:
$$\text{Price of } 1\text{ machine} = 92,465 \div 5 = \mathbf{\text{Rs. } 18,493}$$
a) The price of $10\text{ such washing machines}$:
$$\text{Price} = 18,493 \times 10 = \mathbf{\text{Rs. } 184,930}$$
b) The price of $24\text{ such washing machines}$:
$$\text{Price} = 18,493 \times 24 = \mathbf{\text{Rs. } 443,832}$$
📚 Unit 6 Key Vocabulary & Summary
Pro Tip for Exams: Always write the given quantities clearly with units ($\text{Rs.}$, $\text{kg}$, $\text{km}$, $\text{items}$). State the value of $1\text{ unit}$ clearly before computing the target answer!
More Chapter Notes for Class 5 (FBISE)
MathematicsTest Your Knowledge on Chapter 6: Mastery Guide: The Unitary Method, Direct Proportion, Rates & Financial Applications
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Class 5 Mathematics - Ch 6: Unitary Method Chapter Mock Test
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