Chapter 1 · Practice set 4Textbook page 18

Practice Set 4 – Animated Solutions ചിന്തിക്കാം

The seven questions at the end of the chapter mix everything: angles, division with remainders, two-digit numbers, prices and motion. The recipe never changes – name the two unknowns, turn each fact into an equation, then eliminate one letter. Try each question first, use the hint if you need it, and then watch the animation.

📖 Tools you will use (pages 4, 7 and 9 of this guide)

Question 1 – angles of a right triangle

Q1The difference of the two smaller angles of a right triangle is 20°. Find all three angles.
(ഒരു മട്ടത്രികോണത്തിലെ ചെറിയ കോണുകളുടെ വ്യത്യാസം 20°. അതിന്റെ കോണുകൾ മൂന്നും കണക്കാക്കുക.)

The right angle takes 90° of the 180°, so the two smaller angles add to 90°. Now you know their sum and their difference.

🎬 Solution animation
Answer: x + y = 90 and x − y = 20 give 2x = 110, x = 55, y = 35. The angles are 90°, 55° and 35°.
🌍 Why this matters

A set square, a ramp or a roof truss: in every right triangle the two other angles share 90° between them. One more fact fixes both.

Question 2 – quotient and remainder

Q2When the larger of two numbers is divided by the smaller, the quotient and the remainder are both 2. When 5 times the smaller is divided by the larger, the quotient and remainder are again both 2. Find the numbers.

Dividend = divisor × quotient + remainder. So L = 2S + 2 and 5S = 2L + 2.

🎬 Solution animation
Answer: L = 2S + 2; then 5S = 2(2S + 2) + 2 = 4S + 6, so S = 6 and L = 14. The numbers are 14 and 6. Check: 14 ÷ 6 = 2 remainder 2 ✓; 30 ÷ 14 = 2 remainder 2 ✓
💡 Note

Substitution is the natural method here, because the first fact already says what L is.

Question 3 – swapping the digits

Q3The sum of the digits of a two-digit number is 11. If the digits are interchanged, the new number is 27 more than the original. What is the number?

Tens digit a, units digit b. The number is 10a + b and the reversed number 10b + a. Their difference is 9(b − a).

🎬 Solution animation – digit cards
Answer: a + b = 11 and (10b + a) − (10a + b) = 9(b − a) = 27, so b − a = 3. Then b = 7, a = 4: the number is 47. Check: 74 − 47 = 27 ✓
🌍 Why this matters

Reversing the digits always changes a number by a multiple of 9 – a fact used in old book-keeping to spot two digits typed in the wrong order.

Question 4 – trophies and medals

Q417 trophies and 16 medals cost ₹2180. 16 trophies and 17 medals cost ₹2110. What is the price of a trophy? Of a medal?

The 16 and 17 are swapped. Add the two bills, then subtract them.

🎬 Solution animation – add and subtract
Answer: 17t + 16m = 2180, 16t + 17m = 2110. Adding: 33(t + m) = 4290, so t + m = 130. Subtracting: t − m = 70. So a trophy costs ₹100 and a medal ₹30. Check: 1700 + 480 = 2180 ✓
💡 Note

Multiplying by 17 and 16 would work too, but with big numbers. Spotting the swap saves a lot of arithmetic.

Question 5 – distance with speeding up

Q5An object starts with speed u m/s and its speed increases by a m/s every second, moving in a straight line. In t seconds it travels ut + ½at² metres. It travels 10 m in 2 seconds and 28 m in 4 seconds. What was its starting speed, and by how much does its speed increase every second?

Put t = 2: 2u + 2a = 10. Put t = 4: 4u + 8a = 28. Halve both and subtract.

🎬 Solution animation
Answer: 2u + 2a = 10 ⇒ u + a = 5; 4u + 8a = 28 ⇒ u + 2a = 7. Subtracting: a = 2, then u = 3. Starting speed 3 m/s; the speed increases by 2 m/s every second. Check: 3 × 4 + ½ × 2 × 16 = 12 + 16 = 28 ✓
🌍 Why this matters

Engineers find a vehicle's starting speed and acceleration exactly like this – from two distance readings taken at two times.

Question 6 – six times the digit sum

Q6A two-digit number is 6 times the sum of its digits. The number got by interchanging the digits is 9 more than 4 times the sum of the digits. What is the number?

10a + b = 6(a + b) simplifies to 4a = 5b. The second fact, 10b + a = 4(a + b) + 9, simplifies to 2b − a = 3.

🎬 Solution animation
Answer: 4a = 5b and 2b − a = 3 ⇒ a = 2b − 3 ⇒ 4(2b − 3) = 5b ⇒ 3b = 12 ⇒ b = 4, a = 5. The number is 54. Check: digit sum 9, 6 × 9 = 54 ✓; 45 = 4 × 9 + 9 ✓
💡 Note

First tidy each equation (collect the a's and b's on one side). Messy equations hide simple facts like 4a = 5b.

Question 7 – add 11, add 20

Q7Of two numbers, adding 11 to the first gives twice the second, and adding 20 to the second gives twice the first. Find the numbers.

x + 11 = 2y and y + 20 = 2x. From the first, x = 2y − 11; substitute into the second.

🎬 Solution animation
Answer: x = 2y − 11, so y + 20 = 2(2y − 11) = 4y − 22, giving 3y = 42, y = 14 and x = 17. The numbers are 17 and 14. Check: 17 + 11 = 28 = 2 × 14 ✓; 14 + 20 = 34 = 2 × 17 ✓
🌍 Why this matters

Puzzles where each quantity is described through the other are exactly what substitution was made for.

Check yourself

🧭 Summary

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