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Lesson 11 of 13

Finding the Unknown · Lesson 11 of 13

Equations from Diagrams and Machines

“Read pictures and operation routes carefully before building an equation.”

Learning Objectives

• Translate hidden groups and equal note counts into equations. • Distinguish a serial machine from two parallel branches. • Model a fixed fee and a distance charge. • Use length and angle totals to determine unknown measurements.

A picture hides a quantity, not its relationships

A blob covering several dots can represent an unknown count. If matching blobs hide the same number of dots, use the same letter for each. Visible dots are ordinary known numbers. Count groups and visible objects carefully before equating the total: the equation must describe what the picture actually shows.

bbbThree equal hidden groups + four visible dots = 25
Equal blobs hide equal counts— Each b is the hidden count in one blob; four dots remain visible.
Find what one blob covers

Problem
Three identical blobs hide dots, and four more dots are visible. There are 25 dots in total. How many dots does one blob hide?

  1. 1.Let b be the number hidden by one blob. The total gives 3b + 4 = 25.
  2. 2.Subtract 4: 3b = 21. Divide by 3: b = 7 dots.
  3. 3.Check: 3 × 7 + 4 = 25. The hidden total is 21, while the count under one blob is 7.
Equal counts do not mean equal money

Problem
Ranju has ₹750 in ₹50 and ₹100 notes, with equal numbers of both kinds. Find the note counts.

  1. 1.Let n be the number of each kind. The money total is 50n + 100n = 750.
  2. 2.Combine terms: 150n = 750. Divide by 150: n = 5.
  3. 3.There are five ₹50 notes and five ₹100 notes, ten notes altogether.
  4. 4.Check: 5 × 50 + 5 × 100 = ₹750. The two kinds have equal counts but different total values.

Serial machines: one output becomes the next input

In a serial machine, the number passes through operations in order. The output of one box is fed into the next. Write each intermediate expression before simplifying it. This preserves the route and makes reversing the machine straightforward: undo the final box first.

InputxAdd 3x + 3Multiply by 44(x + 3)Subtract 54(x + 3) − 5For input 12: 12 → 15 → 60 → 55
Follow a serial route— Add 3 first, then multiply the entire result by 4, then subtract 5.
Recover a serial input

Problem
A machine adds 3, multiplies by 4, then subtracts 5. What inputs give outputs 43 and 75?

  1. 1.For output 43, write 4(x + 3) − 5 = 43. Add 5: 4(x + 3) = 48.
  2. 2.Divide by 4: x + 3 = 12. Subtract 3: x = 9. Check: 9 → 12 → 48 → 43.
  3. 3.For output 75, reverse directly: add 5 to get 80, divide by 4 to get 20, subtract 3 to get 17.
  4. 4.Check: 17 → 20 → 80 → 75. Both inputs satisfy the same machine rule.
Repeated operations still need separate undoing

Problem
One machine divides by 3 twice and gives 5. Another subtracts 4 twice and gives −11. Find each input.

  1. 1.For the first, x/3/3 = 5. Reverse the last division: 5 × 3 = 15. Reverse the earlier division: 15 × 3 = 45.
  2. 2.Check: 45 ÷ 3 = 15; 15 ÷ 3 = 5.
  3. 3.For the second, x − 4 − 4 = −11. Reverse the last subtraction: −11 + 4 = −7; reverse the earlier subtraction: −7 + 4 = −3.
  4. 4.Check: −3 − 4 = −7; −7 − 4 = −11.

Parallel machines: the same input enters two branches

A different machine sends the same starting number down two routes. The top route multiplies by 3; the bottom route adds 3. The final box subtracts the bottom result from the top result. Here +3 is not followed by ×3: these operations happen separately. For input 12, the two branch outputs are 36 and 15, and their difference is 21.

InputxMultiply by 33xAdd 3x + 3Top − bottom3x − (x + 3)Both routes receive the original x.
Parallel routes meet at subtraction— The entire lower output x + 3 is subtracted; this needs brackets.
Find a parallel input

Problem
Find the input when the parallel machine gives output 63, then output 227.

  1. 1.The rule is 3x − (x + 3). For 63, write 3x − (x + 3) = 63.
  2. 2.Remove brackets carefully: 3x − x − 3 = 63, so 2x − 3 = 63. Add 3: 2x = 66; divide by 2: x = 33.
  3. 3.Check the branches: top = 99, bottom = 36, difference = 63.
  4. 4.For 227, write 2x − 3 = 227. Add 3: 2x = 230; divide by 2: x = 115.
  5. 5.Check: top = 345, bottom = 118, difference = 227.
Common mistake

A branched route is not a single chain. Also, subtracting x + 3 gives −x − 3, not −x + 3. The brackets specify that the entire bottom output is removed.

Separate fixed amounts from repeated amounts

A taxi fare can contain a fixed hiring charge and an amount per kilometre. A length diagram can likewise contain fixed-width pieces and repeated equal gaps. In both cases, add the fixed amounts and multiply the repeated amount by how many times it occurs. Attach units to your unknown so that the final answer has a clear meaning.

A taxi bill

Problem
A taxi charges ₹800 plus ₹20 per kilometre. The total is ₹2200. Find the distance.

  1. 1.Let d be the distance in kilometres. Write 800 + 20d = 2200.
  2. 2.Subtract 800: 20d = 1400. Divide by 20: d = 70 km.
  3. 3.Check: 800 + 20 × 70 = ₹2200. Dividing the entire bill by 20 would wrongly include the fixed charge as distance cost.
Frame: 3 cmGap: gRod: 2 cmGap: gRod: 2 cmGap: gRod: 2 cmGap: gRod: 2 cmGap: gFrame: 3 cm34 cm2 frames + 4 rods + 5 equal gaps
Count all vertical pieces of a grill— The full height is 34 cm. Both frames are 3 cm, all four rods are 2 cm, and there are five equal gaps. Schematic.
Find an equal gap

Problem
A window grill is 34 cm high, with top and bottom frames of 3 cm each, four rods of thickness 2 cm each, and five equal clear gaps. Find a gap.

  1. 1.Let g be the gap in centimetres. The full height gives 2 × 3 + 4 × 2 + 5g = 34.
  2. 2.The frames and rods use 6 + 8 = 14 cm, so 14 + 5g = 34.
  3. 3.Subtract 14: 5g = 20. Divide by 5: g = 4 cm.
  4. 4.Check: 6 + 8 + 5 × 4 = 34 cm. Count five spaces, not four, because spaces occur above and below the rods too.

Angle sums turn labels into an equation

A triangle’s three interior angles add to 180°. In an isosceles triangle, the two equal sides have equal opposite angles. A side equality shown by matching short strokes therefore supplies angle information even when one base angle is not labelled. Use the given relationships to express all three angles before adding them.

yy + 15°y + 15°Equal sides → equal base anglesxx − 10°x + 10°All three angles total 180°
Use every angle relationship— Matching side strokes in the first triangle justify the repeated base angle. Diagrams are schematic.
Equal base angles

Problem
An isosceles triangle has top angle y° and one base angle (y + 15)°. Find all three angles.

  1. 1.The other base angle is also (y + 15)° because the opposite sides are equal.
  2. 2.Write y + (y + 15) + (y + 15) = 180. Combine: 3y + 30 = 180.
  3. 3.Subtract 30: 3y = 150. Divide by 3: y = 50.
  4. 4.The angles are 50°, 65°, and 65°; they total 180° and the base angles agree.
Three connected angle labels

Problem
A triangle has angles x°, (x − 10)°, and (x + 10)°. Find them.

  1. 1.Add the three labels: x + (x − 10) + (x + 10) = 180.
  2. 2.The −10 and +10 cancel, giving 3x = 180. Divide by 3: x = 60.
  3. 3.The angles are 60°, 50°, and 70°. Check their sum: 60 + 50 + 70 = 180°.
A missing number inside nested operations

Problem
Find x if 37 − (33 − x) = 35.

  1. 1.Subtract 37 from both sides: −(33 − x) = −2.
  2. 2.Multiply both sides by −1: 33 − x = 2. Subtract 33: −x = −31.
  3. 3.Multiply by −1: x = 31. Check: 37 − (33 − 31) = 37 − 2 = 35.

Quiz

Quick check

Three equal blobs and four visible dots total 25. How many dots does one blob hide?

Quick check

Ranju has five ₹50 notes and five ₹100 notes. How many notes are there altogether?

Quick check

For the parallel machine, which expression means top 3x minus bottom x + 3?

Quick check

Four rods between two frames create how many clear gaps?

Quick check

The isosceles triangle has top angle 50°. What are its two equal base angles?

Quick check

An input is divided by 3 twice to give 5. What was the input?

Practice Problems

Practice Problems
  1. Find the missing values in 5x − 8 = 37 and −3(−11 + x) = 45. Check both originals.
  2. Find the input of the serial add-3, multiply-by-4, subtract-5 machine if its output is 99.
  3. Find the input of the parallel 3x minus (x + 3) machine if its output is 41. Calculate both branch outputs.
  4. A taxi charges ₹600 plus ₹25 per kilometre. The bill is ₹1850. Find the distance.
  5. A 46 cm grill has two 3 cm frames, four 2 cm rods, and five equal gaps. Find a gap.
  6. An isosceles triangle has top angle a° and each base angle (a + 30)°. Find all angles.
  7. A triangle has angles (b − 20)°, b°, and (b + 20)°. Find b and each angle.
  8. Explain why equal counts of ₹50 and ₹100 notes should be modelled by 50n + 100n rather than by equal money totals.

Key Takeaways

Key Takeaways

• Count equal groups and visible quantities before writing an equation. • Serial operations are followed in order and reversed in reverse order. • Parallel branches receive the same original input. • Fixed fees, frame widths, and repeated amounts must be counted separately. • Triangle angle sums and equal-side information supply equations for unknown angles.

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A Pinch of History

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Connected Word Problems and Challenges