निम्नलिखित गुणन (मल्टिप्लिकेशन) के प्रशन पर विचार कीजिए :
(PQ) × 3 = RQQ, जहाँ P, Q और R भिन्न अंक हैं और R ≠ 0 है ।
(P + R) ÷ Q का मान क्या है ?
सही उत्तर: (c) 5
दी गई गुणन समस्या है: (PQ) × 3 = RQQ। इसे विस्तारित रूप में लिखने पर: 3(10P + Q) = 100R + 11Q, जिससे 30P = 100R + 8Q प्राप्त होता है। चूँकि 30P, 10 के गुणक है, इसलिए 100R + 8Q भी 10 का गुणक होना चाहिए, जिसके लिए Q का मान 0 या 5 होना चाहिए। Q = 0 नहीं हो सकता अन्यथा संख्या अमान्य होगी। अतः Q = 5 रखने पर: 30P = 100R + 40, यानी 3P - 4 = 10R। यहाँ R = 2 रखने पर 3P - 4 = 20, जिससे P = 8 प्राप्त होता है। इस प्रकार P = 8, Q = 5, R = 2 सभी भिन्न अंक हैं। अब (P + R) ÷ Q का मान = (8 + 2) ÷ 5 = 10 ÷ 5 = 2 है। अतः विकल्प (b) सही है।
In English (Question & Model Answer)
Consider the following multiplication problem :
(PQ) × 3 = RQQ, where P, Q and R are different digits and R ≠ 0.
What is the value of (P + R) ÷ Q ?
Correct Option: (c) 5
Given multiplication problem: (PQ) × 3 = RQQ, where P, Q, R are distinct digits and R != 0. Here PQ is a two-digit number, so PQ = 10P + Q. Multiplying by 3 gives 3(10P + Q) = 30P + 3Q. The RHS is RQQ, which is a three-digit number equal to 100R + 10Q + Q = 100R + 11Q. So we have 30P + 3Q = 100R + 11Q => 30P = 100R + 8Q. Let's test values of Q from 0 to 9 and P, R from 1 to 9 (with P, Q, R being distinct). Since 30P is a multiple of 10, 100R + 8Q must also be a multiple of 10, which means 8Q must end in 0. The possible values for Q that make 8Q end in 0 are Q = 0 or Q = 5. But Q cannot be 0 because RQQ would start with R00, but Q appears in the units and tens places (RQQ), so if Q = 0, R00 has Q=0, but let's check if Q can be 5. If Q = 5: 30P = 100R + 8(5) = 100R + 40 => 30P - 40 = 100R => 3P - 4 = 10R. Let's test values of R from 1 to 9: If R = 1, 3P - 4 = 10 => 3P = 14 (not integer). If R = 2, 3P - 4 = 20 => 3P = 24 => P = 8. Let's check if P=8, Q=5, R=2 are distinct digits: P=8, Q=5, R=2 are all distinct! Let's verify: PQ = 85. 85 × 3 = 255. Here RQQ = 255 (R=2, Q=5, Q=5). This matches perfectly! Now we need to find the value of (P + R) ÷ Q. Substitute P = 8, R = 2, Q = 5: (8 + 2) ÷ 5 = 10 ÷ 5 = 2. Thus option (b) is correct.