Unitary Method

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ऐकिक नियम (Unitary Method)

अंकगणितमा Unitary Method भनेको big idea हो। Unitary Method सँग सम्बन्धित विद्यालयका पाठ्यक्रमहरूले तीनचटा मुख्य कुराहरू: भिन्न, अनुपात, र प्रतिशत समेटेको हुन्छ।

1. पहिलो कुरा, Unitary Method ले धेरै समस्याहरूलाई mental algorithms रूपमा हल गर्न मद्दत गर्छ र यो mental arithmetic को महत्वपूर्ण भाग हो। यसलाई एकपटक राम्रोसँग बुझिसकेपछि, दैनिक जीवनका बिविध परिस्थितिहरूमा छिटो र सहज रूपमा समस्या समाधान गर्न सकिन्छ।
2. दोस्रो, Unitary Method मा दक्षता हासिल गर्नुले भिन्न, अनुपात, र प्रतिशत संख्याको संरचनालाई राम्रोसँग बुझ्न मद्दत गर्छ, जसमा अंश (numerator) र हर (denominator) र यस्तै बिभिन्न रुपको को स्पष्ट सम्बन्ध हुन्छ।

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THE BASICS of UNITARY METHOD

Although unitary method is a mental algorithm, it should be taught using three successive parallel sentences. The first sentence restates the problem, and the last states its solution. For example

If 4 mangoes cost 120, how much do 9 mangoes cost?

	4	mangoes cost	=120
÷ 4	1	mango costs	\(=\dfrac{120}{4}=30\)
× 9	9	mangoes cost	\(=30 \times 9=270\)

The method relies on a sequence of parallel sentences. Once the method is mastered, the successive sentences in easier examples can merely be spoken or thought.

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Definition

The method of first finding the value of one article/unit and then, finding the value of more articles/units is called Unitary Method.

In Unitary Method, we use three types of proportions.

1. Direct Proportion (सीधा अनुपात)
   दुईवटा परिमाणहरु मध्ये यदि एक परिमाण बढदा/घट्दा, अर्को परिमाण पनि समानुपातिक रूपमा बढ्छ/घट्छ भने त्यस्तो परिमाणहरुलाई Direct Proportion भनिन्छ। जस्तै बस्तुको सङ्ख्या बढाइयो भने, लाग्ने मुल्य पनि बढ्छ। त्यसैले बस्तुको सङ्ख्या र मुल्य Direct Proportion भएको परिमाणहरु हुन्।

   Weight (kg) \(\uparrow\)	Cost (Rs.) \(\uparrow\)
   10	2,750
   15	x

   The ratio is
   \(\dfrac{x}{2750} =\dfrac{15}{10} \)
   
   
2. Inverse Proportion (प्रतिलोम अनुपात)
   दुईवटा परिमाणहरु मध्ये यदि एक परिमाण बढदा/घट्दा, अर्को परिमाण पनि समानुपातिक रूपमा घट्छ/बढ्छ भने त्यस्तो परिमाणहरुलाई Indirect Proportion भनिन्छ। जस्तै एउटा कार्य गर्नका लागि ५ जना मजदुर लाग्छ र उनीहरूले १० दिन मा काम पूरा गर्छन्। यदि मजदुरहरूको सङ्ख्या दोब्बर (१० जना) गरियो भने, सोही कार्य सम्पन्न गर्न लाग्ने समय घट्छ। त्यसैले कामदारको सङ्ख्या र कार्य सम्पन्न गर्न लाग्ने समय Indirect Proportion भएको परिमाणहरु हुन्।

   People\(\uparrow\)	Days\(\downarrow\)
   5	10
   10	x

   The ratio is
   \(\dfrac{x}{10} =\dfrac{5}{10} \)
   
   
3. Chain Rule (शृंखला नियम)
   Chain Rule भनेको UNITARY METHOD को विस्तारित रुप हो जसले धेरै परिमाणहरू आपसमा सम्बन्धित गर्छ। जस्तै यदि A ले कुनै काम १० दिनमा गर्छ, B ले सोही काम २० दिनमा गर्छ, र C ले ३० दिनमा गर्छ भने, A, B, र C मिलेर काम गर्दा कति दिन लाग्छ?

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Drag and Drop Quiz: Q1

बायाँबाट आइटमहरू तान्नुहोस् र तिनीहरूको सही अनुपात प्रकार वा परिभाषामा राख्नुहोस्:

\(\text{Cost} \uparrow \implies \text{Quantity} \uparrow\)

\(\text{Workers} \uparrow \implies \text{Days} \downarrow\)

\(\text{Ratio} \ \frac{A}{B} = k \ (\text{Constant})\)

\(\text{Distance} \uparrow \implies \text{Fuel Used} \uparrow\)

\(\text{Speed} \uparrow \implies \text{Time Taken} \downarrow\)

\(\text{Workers} \times \text{Days} = k\)

\(\text{Weight} \uparrow \implies \text{Cost} \uparrow\)

\(\text{Pipe Diameter} \uparrow \implies \text{Filling Time} \downarrow\)

सीधा अनुपात (Direct Proportion)

प्रतिलोम अनुपात (Inverse Proportion)

सीधा अनुपात सूत्र (Direct Formula)

सीधा अनुपात उदाहरण

प्रतिलोम अनुपात उदाहरण

प्रतिलोम अनुपात सूत्र (Inverse Formula)

सीधा अनुपात उदाहरण

प्रतिलोम अनुपात उदाहरण

Score: 0/10

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Multiple Choice Quiz1: Try it

Question 1 of 5

Question text will appear here

Quiz Complete! Your Score: 0/5

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Questions

1. १५ जना कामदारले एउटा काम २५ दिनमा सक्छन् भने कति कामदार थप्दा सो काम १५ दिनमा सकिन्छ? 15 workers can do a piece of work in 25 days. How many workers should be added to complete the same work in 15 days?
Given that

People\(\uparrow\)	Days\(\downarrow\)
15	25
15+x	15
We know that "number of people" and "number of days" to complete a work have indirect relation (more people → less days), therefore, using the relation indirect relation, we get
\(\dfrac{15+x}{15} =\dfrac{25}{15} \)

or\(15+x=25\)

or\(x=10\)

Thus, 10 more workers should be added to complete the same work in 15 days
2. १० kg स्याउको मूल्य रु. २,७५० पर्छ भने १५ kg स्याउको मूल्य कति पर्ला? If the cost of 10 kg of apples is Rs. 2,750, what will be the cost of 15 kg of apples?
Given that

Weight (kg) \(\uparrow\)	Cost (Rs.) \(\uparrow\)
10	2,750
15	x
We know that weight and cost have a direct relation (more weight → more cost), so we use direct proportion:
\(\dfrac{x}{15} = \dfrac{2750}{10}\)

or\(10x = 2750 \times 15\)

or\(x = \dfrac{41,250}{10} = 4,125\)

Thus, the cost of 15 kg of apples is Rs. 4,125
3. यदि रु. ६० मा ५ वोटा कलम पाइन्छ भने रु. २४० मा कतिवटा कलम पाइन्छ, पत्ता लगाउनुहोस्। If 5 pens can be bought for Rs. 60, how many pens can be bought for Rs. 240?
Given that

Amount (Rs.) \(\uparrow\)	Number of pens \(\uparrow\)
60	5
240	x
We know that "amount" and "number of pens" have direct relation (more pen → more cost), therefore, using direct proportion, we get
\(\dfrac{x}{5} = \dfrac{240}{60}\)

or\(60x = 5 \times 240\)

or\(x = \dfrac{1200}{60} = 20\)

Thus, 20 pens can be bought for Rs. 240.
4. ५ kg सन्तराको रु. १,१२५ पर्छ भने १० kg सन्तराको मूल्य पत्ता लगाउनुहोस्। If the cost of 5 kg of oranges is Rs. 1,125, find the cost of 10 kg of oranges?
Given that

Weight (kg) \(\uparrow\)	Cost (Rs.) \(\uparrow\)
5	1,125
10	x
We know that weight and cost have a direct relation (more weight → more cost), therefore, using direct proportion, we get
\(\dfrac{x}{1125} = \dfrac{10}{5}\)

or\(5x = 1125 \times 10\)

or\(x = \dfrac{11,250}{5} = 2,250\)

Thus, the cost of 10 kg of oranges is Rs. 2,250.
5. १० जना मानिसले एउटा काम १५ दिनमा पूरा गर्न सक्छन् भने ५ जना मानिसले उक्त काम कति दिनमा गर्न सक्छन्? पत्ता लगाउनुहोस्। If 10 men can complete a work in 15 days, how long will 5 men take to complete the work? Find it.
Given that

People \(\downarrow\)	Days \(\uparrow\)
10	15
5	x
We know that "number of people" and "number of days" have an indirect relation (fewer people → more days), therefore, using indirect proportion, we get
\(\dfrac{x}{15} = \dfrac{10}{5}\)

or\(x = \dfrac{10}{5} \times 15\)

or\(x = 30\)

Thus, 5 men will take 30 days to complete the same work.
6. यदि २० जना कामदारले कुनै एउटा काम ४८ दिनमा सक्छन् भने कति दिनमा ३० कामदारले सो काम सक्छन्? If 20 workers can complete a work in 48 days, then in how many days 20 workers will complete the same work?
Given that

People \(\uparrow\)	Days \(\downarrow\)
20	48
30	x
We know that "number of people" and "number of days" have an indirect relation (more people → fewer days), therefore, using indirect proportion, we get
\(\dfrac{x}{48} = \dfrac{20}{30}\)

or\(x = \dfrac{20}{30} \times 48\)

or\(x = 32\) days

Thus, 30 workers will complete the work in \(32\) days .
7. ५ kg स्याउ र ३ दर्जन केराको जम्मा मूल्य रु. ९०० छ र २ kg स्याउको मूल्य रु. २४० पर्छ भने ५ दर्जन केराको मूल्य पत्ता लगाउनुहोस्। The cost of 5 kg apples and 3 dozen banana is Rs. 900. If the cost of 2 kg apples is Rs. 240, find the cost of 5 dozen banana.
First, find the cost of 1 kg apple:
Cost of 2 kg apples = Rs. 240

∴ Cost of 1 kg apple = \(\dfrac{240}{2} = 120\)

∴ Cost of 5 kg apples = \(5 \times 120 = 600\)

Given that:
Cost of (5 kg apples + 3 dozen bananas) = Rs. 900

⇒ 600 + cost of 3 dozen bananas = 900

⇒ Cost of 3 dozen bananas = 900 − 600 = 300

⇒ Cost of 1 dozen bananas = \(\dfrac{300}{3} = 100\)

⇒ Cost of 5 dozen bananas = \(5 \times 100 = 500\)

Thus, the cost of 5 dozen bananas is Rs. 500.
8. १२ जना कामदारले कुनै एउटा काम २० दिनमा गर्न सक्छन्। सोही काम १६ दिनमा सक्नका लागि कति जना कामदार थप्नुपर्ला? 12 workers can complete a piece of work in 20 days. How many workers should be added to complete the work in 16 days?
Given that

People \(\uparrow\)	Days \(\downarrow\)
12	20
12 + x	16
We know that "number of people" and "number of days" have an indirect relation (more people → fewer days), therefore, using indirect proportion, we get
\(\dfrac{12 + x}{12} = \dfrac{20}{16}\)

or\(12 + x = 15\)

or\(x = 3\)

Thus, 3 more workers should be added to complete the work in 16 days.
9. १५ जना कामदारले १० घण्टा प्रतिदिन काम गरी ३० दिनमा पूरा गरे। २५ दिनमा काम पूरा गर्न प्रतिदिन कति घण्टाका दरले काम गर्नुपर्ला? 15 workers can complete a piece of work in 30 days working 10 hours per day. How many hours per day must he work to complete the work in 25 days?
Given that

Days \(\downarrow\)	Hours per day \(\uparrow\)
30	10
25	x
(Number of workers remains the same, so we compare only days and hours per day.)
We know that "number of days" and "hours per day" have an indirect relation (fewer days → more hours per day), therefore, using indirect proportion, we get
\(\dfrac{x}{10} = \dfrac{30}{25}\)

or\(x = 10 \times \dfrac{30}{25}\)

or\(x = 12\)

Thus, they must work 12 hours per day to complete the work in 25 days.
10. १५ जना मानिसले कुनै काम ४० दिनमा गर्न सक्छन्। यदि ५ जना मानिस घटाउँदा सो काम कति दिनमा सकिएला? 15 men can do a piece of work in 80 days. How long will it take to complete the work if 5 men are reduced?
Given that

People \(\downarrow\)	Days \(\uparrow\)
15	80
15 − 5 = 10	x
We know that "number of people" and "number of days" have an indirect relation (fewer people → more days), therefore, using indirect proportion, we get
\(\dfrac{x}{80} = \dfrac{15}{10}\)

or\(x = \dfrac{15}{10} \times 80\)

or\(x = 120\)

Thus, the work will be completed in 120 days if 5 men are reduced.
11. १८ जना कामदारले एउटै काम ९१ दिनमा गर्न सक्छन् भने ३ जना कामदार थप्दा सो काम कति दिनमा सकिएला? 18 workers can do a piece of work in 91 days. How long will it take to complete the work if 3 workers are added?
Given that

People \(\downarrow\)	Days \(\uparrow\)
18	31
18 + 3 = 21	x
We know that "number of people" and "number of days" have an indirect relation (fewer people → more days), therefore, using indirect proportion, we get
\(\dfrac{x}{91} = \dfrac{21}{15}\)

or\(x = \dfrac{21}{15} \times 91\)

or\(x = 78\)

Thus, the work will be completed in 78 days if 3 workers are added.
12. कुनै एक टुक्रा जमिनको \(\frac{१}{३}\) भागको मूल्य रु. ३,६४० पर्छ भने पूरै जमिनको मूल्य कति पर्छ? त्यसै ४ टुक्राको मूल्य कति पर्ला? [Ans: रु. 7,840] If \(\frac{१}{३}\) of a piece of land is cost Rs. 3,540, what is the cost of the whole land? What is the cost of such 4 pieces?
Given that
Cost of \(\dfrac{1}{4}\) of land = Rs. 3,540

orCost of whole land = \(4 \times 3,540 = 14,160\)

orCost of 4 such whole pieces = \(4 \times 14,160 = 56,640\)

Thus, the cost of the whole land is Rs. 14,160 and the cost of 4 such pieces is Rs. 56,640.
13. ४ दर्जन सिसाकलमको मूल्य रु. २६४ पर्छ भने ५० वटा सिसाकलमको मूल्य कति पर्ला? If the price of 4 dozen pencils is Rs. 264, what will be the price of 50 pencils?
Given that
4 dozen pencils = 4 × 12 = 48 pencils


Number of pencils \(\uparrow\)	Cost (Rs.) \(\uparrow\)
48	264
50	x
We know that number of pencils and cost have a direct relation, therefore, using direct proportion, we get
\(\dfrac{x}{264} = \dfrac{50}{48}\)

or\(x = \dfrac{50}{48} \times 264\)

or\(x = 275\)

Thus, the price of 50 pencils is Rs. 275.
14. यदि ४० kg धानको मूल्य रु. २,५०० भए एक क्विन्टल धानको मूल्य पत्ता लगाउनुहोस्। If the cost of 40 kg rice is Rs. 2,500, find the cost of one quintal rice.
We know that:
1 quintal = 100 kg

Given that

Weight (kg) \(\uparrow\)	Cost (Rs.) \(\uparrow\)
40	2,500
100	x
Since weight and cost have a direct relation, using direct proportion, we get
\(\dfrac{x}{2500} = \dfrac{100}{40}\)

or\(x = \dfrac{100}{40} \times 2500\)

or\(x = 6,250\)

Thus, the cost of one quintal (100 kg) rice is Rs. 6,250.
15. यदि ३० वटा सुन्तलाको मूल्य रु. १८० भए २ दर्जन सुन्तलाको मूल्य पत्ता लगाउनुहोस्। [Ans: रु. 144] If the cost of 30 oranges is Rs. 180, find the cost of 2 dozen oranges.
First, note that:
2 dozen oranges = 2 × 12 = 24 oranges

Given that

Number of oranges \(\uparrow\)	Cost (Rs.) \(\uparrow\)
30	180
24	x
Since number of oranges and cost have a direct relation, using direct proportion, we get
\(\dfrac{x}{180} = \dfrac{24}{30}\)

or\(x = \dfrac{24}{30} \times 180\)

or\(x = 144\)

Thus, the cost of 2 dozen oranges is Rs. 144.
16. ८ वोटा टेबुल र २ वोटा कुर्सीको जम्मा मूल्य रु. १३,००० छ। यदि २ वटा टेबुलको मूल्य रु. २,४०० भए १ वटा कुर्सीको मूल्य पत्ता लगाउनुहोस्। Total cost of 5 tables and 7 chairs is Rs. 19,000. If the cost of 2 tables is Rs. 2,400, find the cost of one chair.
Given that:
Cost of 2 tables = Rs. 2,400

∴ Cost of 1 table = \(\dfrac{2400}{2} = 1,200\)

∴ Cost of 8 tables = \(8 \times 1,200 = 9,600\)

Also given:
Cost of (8 tables + 2 chairs) = Rs. 13,000

⇒ 9,600 + cost of 2 chairs = 13,000

⇒ Cost of 2 chairs = 13,000 − 9,600 = 3,400

⇒ Cost of 1 chair = \(\dfrac{3,400}{2} = 1,700\)

Thus, the cost of one chair is Rs. 1,700.
17. एउटा बसलाई ७५ km दूरी पार गर्न १५ लिटर डिजेल चाहिन्छ। २०० लिटर डिजेलले कति km दूरी पार गर्न सक्ला? A bus needs 15 litres of diesel to travel 75 km of distance. How many kilometers will it travel on 200 litres of diesel?
Given that

Diesel (litres) \(\uparrow\)	Distance (km) \(\uparrow\)
15	75
200	x
We know that diesel and distance have a direct relation (more diesel → more distance), therefore, using direct proportion, we get
\(\dfrac{x}{75} = \dfrac{200}{15}\)

or\(x = \dfrac{200}{15} \times 75\)

or\(x = 1,000\)

Thus, the bus will travel 1,000 km on 200 litres of diesel.
18. अनुजाले ४ km/hr को गतिमा साइकल चलाउँदा ७०० मिटर दूरी पार गर्न कति समय लाग्ला? How long will Anuja take to cover 700 meters of distance when she rides a bicycle at the speed of 4 km/hr?
Given:
Speed = 4 km/hr
Distance = 700 metres = \(\dfrac{700}{1000} = 0.7\) km

We know that:
Time = \(\dfrac{\text{Distance}}{\text{Speed}}\)

Time = \(\dfrac{0.7}{4} = 0.175\) hours

Convert hours into minutes:
0.175 × 60 = 10.5 minutes

or10 minutes 30 seconds

Thus, Anuja will take 10 minutes 30 seconds to cover 700 metres.
19. एउटा होस्टलमा ६०० विद्यार्थीको लागि ५० दिनको पुग्ने खाने रसद छ। १५ दिनपछि, १८० जना विद्यार्थीले होस्टल छोडे भने सो खाने कति दिन पुग्छ होला? A hostel has food for 600 students for 50 days. After 15 days, 180 students leave the hostel. How long will the food last?
Given that

Students \(\uparrow\)	Days \(\downarrow\)
600	50-15
600-180	x
We know that "number of students" and "number of days" have an indirect relation (more people → fewer days), therefore, using indirect proportion, we get
\(\dfrac{x}{35} = \dfrac{600}{420}\)

orx = \(x=\dfrac{35 \times 600}{420} = 50\)

Thus, the remaining food will last for 50 more days.

20. १०० जना सिपाहीलाई ६० दिनलाई पुग्ने रसद छ। यदि १२ दिनपछि, २०० जना सिपाहीहरू उनीहरूसँग थपिए भने सो रसद कति समयसम्म चल्ला? 100 soldiers have provision for 60 days at 60 days. If 200 soldiers join them after 12 days, how long will the remaining provisions last?
Given that

Soldiers \(\uparrow\)	Days \(\downarrow\)
100	60 − 12 = 48
100 + 200 = 300	x
We know that "number of soldiers" and "number of days" have an indirect relation (more soldiers → fewer days), therefore, using indirect proportion, we get
\(\dfrac{x}{48} = \dfrac{100}{300}\)

or\(x = \dfrac{100}{300} \times 48 = 16\)

Thus, the remaining provision will last for 16 more days.