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Unitary Method

Practice MCQs

None

Quantitative Aptitude Unitary Method Competitive Exams

Unitary Method is a basic but very powerful quantitative aptitude technique used to find the value of one unit first and then calculate the value of the required number of units. It is widely used in questions on ratio, proportion, percentage, time and work, speed, cost, wages, production, and daily-life arithmetic.

What is Unitary Method?

Unitary Method means finding the value of one unit first, and then using that value to find the value of many units or fewer units.

For example, if 5 pens cost ₹100, then the cost of 1 pen is ₹20. Therefore, the cost of 8 pens is:

\[ 8 \times 20 = 160 \]

Hence, the cost of 8 pens is ₹160.

Quick idea: First find the value of one unit. Then multiply or divide according to the requirement.
Concept Meaning Example
One Unit Value of a single item or quantity Cost of 1 pen
Many Units Value of more than one item Cost of 10 pens
Direct Variation More quantity means more value More books cost more money
Inverse Variation More quantity means less value of another quantity More workers take less time
Unit Rate Rate for one unit ₹20 per pen

“In unitary method, the value of one unit is the bridge to the final answer.”

Aptitude Tip
Key points
  • Find the value of one unit first.
  • Then find the value of required units.
  • Use multiplication for direct variation.
  • Use inverse relation for workers-time type problems.
  • Keep units consistent.
  • Useful in cost, quantity, time, work, and speed questions.
one unit unit rate direct inverse proportion

Visual Understanding

These diagrams show how unitary method converts a given quantity into one unit and then into the required quantity.

Finding One Unit
5 pens cost ₹100 Pen Pen Pen Pen Pen 1 pen = ₹20
\[ \text{Cost of 1 pen}=\frac{100}{5}=20 \]

Divide total value by total units to get the value of one unit.

Finding Required Units
1 pen ₹20 × 8 8 pens ₹160 Once one-unit value is known, multiply by required units
\[ \text{Cost of 8 pens}=20\times8=160 \]

Multiply the one-unit value by the required number of units.

Direct Variation
2 units 4 units 6 units More units → More value
\[ \text{Direct variation: } x \uparrow \Rightarrow y \uparrow \]

Cost, quantity, distance, and production often follow direct variation.

Inverse Variation
More workers → Less time 2 workers 12 days 4 workers 6 days When one quantity doubles, the other may become half
\[ \text{Inverse variation: } x \uparrow \Rightarrow y \downarrow \]

Workers and time often follow inverse variation when total work is fixed.

Important Formulas and Rules

One Unit Value
\[ \text{One unit}=\frac{\text{Total value}}{\text{Total units}} \]

First step of unitary method.

Required Units Value
\[ \text{Required value}=\text{One unit value}\times\text{Required units} \]

Used after finding one-unit value.

Direct Proportion
\[ x_1:y_1=x_2:y_2 \]

More quantity gives more value.

Inverse Proportion
\[ x_1y_1=x_2y_2 \]

More of one quantity gives less of another.

Cost of Items
\[ \text{Cost per item}=\frac{\text{Total cost}}{\text{Number of items}} \]

Used in buying and selling questions.

Work and Workers
\[ M_1D_1=M_2D_2 \]

Used when total work and efficiency are same.

Speed and Time
\[ S_1T_1=S_2T_2 \]

For fixed distance, speed and time are inverse.

Quantity and Cost
\[ \frac{Q_1}{C_1}=\frac{Q_2}{C_2} \]

For fixed rate, quantity and cost are direct.

Rule: In direct variation, multiply after finding one unit. In inverse variation, remember that increasing one quantity reduces the other when total work or distance is fixed.

Common Types of Questions

Cost and Quantity

Find cost of required items from given cost of some items.

  • Cost of books
  • Cost of fruits
  • Price per kg
  • Quantity from amount
Work and Workers

Find number of workers or days needed for fixed work.

  • More workers
  • Fewer workers
  • Days required
  • Work completion
Speed and Time

For fixed distance, speed and time are inversely related.

  • Speed increased
  • Time reduced
  • Journey time
  • Fixed distance
Production and Consumption

Questions based on output, usage, and daily consumption.

  • Factory production
  • Food consumption
  • Daily output
  • Stock duration
Exam approach: Identify whether the relation is direct or inverse before solving. This decision is the most important step in unitary method.
Method Bank
Cost of Items

5 pens cost ₹100.

\[ 1\text{ pen}=\frac{100}{5}=20 \]
Required Cost

Cost of 8 pens.

\[ 8\text{ pens}=20\times8=160 \]
Workers and Days

4 workers take 12 days.

\[ 8\text{ workers take } \frac{4\times12}{8}=6\text{ days} \]
Speed and Time

Speed doubles for same distance.

\[ \text{Time becomes half} \]

Tip: In direct cases, more units mean more value. In inverse cases, more workers or speed means less time.

Unitary Method Solving Flow
Given Units Find 1 Unit Required Units Final Answer Given value → one unit → required value
This is the basic solving flow for most direct unitary method questions.
\[ \text{One unit value}=\frac{\text{Given value}}{\text{Given units}} \]
\[ \text{Required value}=\text{One unit value}\times\text{Required units} \]

Step-by-Step Solving Method

Step Direct Variation Inverse Variation
Step 1 Identify the given units and given value. Identify the given quantity and time/value.
Step 2 Find the value of one unit. Find total work or constant product.
Step 3 Multiply by required units. Use inverse relation to find required value.
Step 4 Check the unit of answer. Check whether value should increase or decrease.
Step 5 Write final answer clearly. Write final answer with correct unit.
Important: Before calculation, ask: “If quantity increases, should the answer increase or decrease?” This helps decide direct or inverse variation.

Solved Examples

Question Method Answer
If 5 pens cost ₹100, find the cost of 8 pens. Cost of 1 pen:
\[ \frac{100}{5}=20 \]
Cost of 8 pens:
\[ 20\times8=160 \]
₹160
If 12 books cost ₹600, find the cost of 7 books. Cost of 1 book:
\[ \frac{600}{12}=50 \]
Cost of 7 books:
\[ 50\times7=350 \]
₹350
If 4 kg rice costs ₹240, find the cost of 9 kg rice. Cost of 1 kg:
\[ \frac{240}{4}=60 \]
Cost of 9 kg:
\[ 60\times9=540 \]
₹540
If 8 workers complete a work in 15 days, how many days will 12 workers take? This is inverse variation.
\[ M_1D_1=M_2D_2 \]
\[ 8\times15=12\times D_2 \]
\[ D_2=\frac{8\times15}{12}=10 \]
10 days
If a car travels 180 km in 3 hours, how far will it travel in 5 hours at the same speed? Distance in 1 hour:
\[ \frac{180}{3}=60\text{ km} \]
Distance in 5 hours:
\[ 60\times5=300\text{ km} \]
300 km
If 6 machines produce 900 items in a day, how many items will 10 machines produce? Production by 1 machine:
\[ \frac{900}{6}=150 \]
Production by 10 machines:
\[ 150\times10=1500 \]
1500 items
If 3 men can do a work in 20 days, how many men are required to complete it in 12 days? This is inverse variation.
\[ M_1D_1=M_2D_2 \]
\[ 3\times20=M_2\times12 \]
\[ M_2=\frac{60}{12}=5 \]
5 men
If 10 packets cost ₹250, how many packets can be bought for ₹400? Cost of 1 packet:
\[ \frac{250}{10}=25 \]
Number of packets:
\[ \frac{400}{25}=16 \]
16 packets

Note: In unitary method, always decide whether the situation is direct or inverse before solving.

Common Traps and Shortcuts

Common Traps
  • Using direct variation when the relation is inverse.
  • Forgetting to find one-unit value first.
  • Mixing units such as kg and grams.
  • Multiplying instead of dividing to find one unit.
  • Ignoring whether the answer should increase or decrease.
  • Not writing the correct unit in the final answer.
Useful Shortcuts
  • For direct cases, find one unit and multiply.
  • For inverse cases, use constant product.
  • More items means more cost.
  • More workers means fewer days.
  • More speed means less time for same distance.
  • Always check final answer logically.
Exam approach: Ask whether the answer should become bigger or smaller. This helps identify the correct relationship quickly.

Practice

A) Multiple Choice Questions
  1. If 4 pencils cost ₹40, find the cost of 9 pencils.
    ₹60 ₹80 ₹90 ₹100
  2. If 6 kg sugar costs ₹300, find the cost of 1 kg sugar.
    ₹40 ₹50 ₹60 ₹70
  3. If 5 workers complete a work in 20 days, 10 workers will complete it in:
    5 days 10 days 15 days 40 days
  4. If 3 notebooks cost ₹90, how many notebooks can be bought for ₹210?
    5 6 7 8
  5. In unitary method, the first step is usually to find:
    Total profit One-unit value Percentage increase Compound amount
B) Solve the Higher-Order Problems
  1. If 7 chairs cost ₹3500, find the cost of 12 chairs. (Hint: First find cost of 1 chair.)
  2. If 9 workers complete a work in 16 days, how many days will 12 workers take? (Hint: Workers and days are inversely proportional.)
  3. If a car covers 240 km in 4 hours, how far will it cover in 7 hours at the same speed? (Hint: Find distance covered in 1 hour.)
  4. If 15 packets cost ₹600, how many packets can be bought for ₹1000? (Hint: Find cost of 1 packet first.)
  5. If 5 machines produce 1200 items, how many items will 8 machines produce at the same rate? (Hint: Production is directly proportional to machines.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Unitary Method Finding one-unit value first
Unit Rate Value per single unit
Direct Variation Both quantities increase or decrease together
Inverse Variation One quantity increases while the other decreases
Workers and Days Usually inverse relation
Cost and Quantity Usually direct relation
Aptitude Reminder

The unitary method is useful whenever a question gives the value of many units and asks for one unit or another number of units. Identify whether the relation is direct or inverse before solving.

Task: Create five questions using cost, quantity, workers, time, speed, and production.

Show Suggested Answers
Multiple Choice
  1. ₹90
    Cost of 1 pencil:
    \[ \frac{40}{4}=10 \]
    Cost of 9 pencils:
    \[ 10\times9=90 \]
  2. ₹50
    \[ \text{Cost of 1 kg}=\frac{300}{6}=50 \]
  3. 10 days
    Workers and days are inversely proportional.
    \[ 5\times20=10\times D \]
    \[ D=10 \]
  4. 7
    Cost of 1 notebook:
    \[ \frac{90}{3}=30 \]
    Number of notebooks:
    \[ \frac{210}{30}=7 \]
  5. One-unit value
    In unitary method, we normally find the value of one unit first.
Higher-Order Problems
  1. Cost of 1 chair:
    \[ \frac{3500}{7}=500 \]
    Cost of 12 chairs:
    \[ 500\times12=6000 \]
    Answer = ₹6000.
  2. Workers and days are inverse:
    \[ 9\times16=12\times D \]
    \[ D=\frac{144}{12}=12 \]
    Answer = 12 days.
  3. Distance in 1 hour:
    \[ \frac{240}{4}=60\text{ km} \]
    Distance in 7 hours:
    \[ 60\times7=420\text{ km} \]
    Answer = 420 km.
  4. Cost of 1 packet:
    \[ \frac{600}{15}=40 \]
    Packets for ₹1000:
    \[ \frac{1000}{40}=25 \]
    Answer = 25 packets.
  5. Production by 1 machine:
    \[ \frac{1200}{5}=240 \]
    Production by 8 machines:
    \[ 240\times8=1920 \]
    Answer = 1920 items.
Concept Matching
  1. Unitary Method → Finding one-unit value first
  2. Unit Rate → Value per single unit
  3. Direct Variation → Both quantities increase or decrease together
  4. Inverse Variation → One quantity increases while the other decreases
  5. Workers and Days → Usually inverse relation
  6. Cost and Quantity → Usually direct relation
Clue Explanation

Unitary method is based on finding one-unit value. For cost and quantity, the relation is usually direct. For workers and days or speed and time, the relation is usually inverse.

Exam tips
  • Find one-unit value first.
  • Check whether relation is direct or inverse.
  • Cost and quantity are usually direct.
  • Workers and days are usually inverse.
  • Speed and time are inverse for fixed distance.
  • Always check whether the answer is logically bigger or smaller.
Practice MCQs