A plan is drawn at 1 cm to 2 m. You need the same room at 1 cm to 4 m.
Step 1: Picture It
Start with something you already know.
Pacing out a room
- You pace across a room, and each of your steps is 3 feet.
- It takes you 8 steps, so the room is 24 feet across.
- Your little brother takes steps of 2 feet.
- He needs 24 ÷ 2 = 12 steps for the same room.
The room never changed; only the size of each step did.
On a plan, each centimeter is a step, and the scale is its length.
So redraw through the room: old steps to real feet, then real feet to new steps.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A sports hall plan is drawn at 1 cm to 3 m. On the plan the hall is 7 cm long and 4 cm wide. How long is the real hall, in meters?
meters
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Go through the real length
Convert the old drawing to the real size first, then convert that down to the new drawing.
With those numbers
6 cm at 1 : 2 is 12 m. At 1 cm to 4 m, 12 m becomes 3 cm.
Does that make sense
Each centimeter now covers twice as much ground, so the drawing is half the size. It should shrink.
Step 3: Try It Yourself
Tap and try it out.
Before: 6
After: 3
The factor is less than 1, so the bar got shorter. Multiplying does not always make things bigger.
Step 4: In Real Life
An architect’s drawings
An architect draws a house at 1 to 100, then redraws the kitchen at 1 to 20 so the cabinets show. Same house, different size of picture.
Step 5: Watch an Example
One step at a time.
Watch Ana Redraw a Park Path
A park plan is drawn at 1 cm to 5 m. The main path is 8 cm long on it. Ana redraws the path at 1 cm to 2 m.
- Step 1
Ana names the direction before she moves. She is going from the plan up to the ground, and that multiplies.
Watch Dev Redraw a School Corridor
A school corridor is 12 cm long on a plan at 1 cm to 2 m. Dev redraws it at 1 cm to 8 m. He starts, and you finish.
- Step 1
Dev writes both scales down first. The old plan gives 1 cm to 2 m and the new plan gives 1 cm to 8 m. The corridor is the same corridor in both.
Two Roads to the Same Answer
A playground is 9 cm wide on a plan at 1 cm to 4 m. Maya redraws it at 1 cm to 2 m. There are two roads, and both are right.
- Step 1
Maya's first road goes through the ground. 9 × 4 = 36 m, and then 36 ÷ 2 = 18 cm.
Step 6: Your Turn
Practice makes it stick.
The Garden Path
Problem 1 of 15
Maya's garden plan is drawn at 1 cm to 4 m. The path along the bottom of the garden is 6 cm long on the plan. Gravel costs $9 for every meter of path. What does the gravel cost?
dollars
The New Plan
Problem 2 of 15
A school plan is drawn at 1 cm to 6 m. The hall's long wall is 7 cm on it. The builder wants that wall redrawn at 1 cm to 3 m. The print shop charges $2 for every centimeter of wall drawn on the new plan. What does that wall cost to print?
dollars
Old Plan, Ground, New Plan
Plan up to the ground multiplies. Ground down to a plan divides.
Problem 3 of 15
A fence is 9 cm long on a plan at 1 cm to 4 m. How long is the real fence, in meters?
meters
Problem 4 of 15
A real running track is 63 m long. On a plan at 1 cm to 7 m, how long is it, in cm?
cm
Problem 5 of 15
A path is 8 cm on a plan at 1 cm to 3 m. The plan is redrawn at 1 cm to 4 m. How long is the path on the new plan, in cm?
cm
Problem 6 of 15
A gym is 5 cm long on a plan at 1 cm to 6 m. The plan is redrawn at 1 cm to 2 m. How long is the gym on the new plan, in cm?
cm
Problem 7 of 15
A plan at 1 cm to 20 m is redrawn at 1 cm to 5 m. What single number does every length on the drawing get multiplied by?
Whose Step?
One line in each piece of working is a slip. Find it, then fix it.
Problem 8 of 15
Noor redrew a 10 cm wall from a plan at 1 cm to 4 m onto a plan at 1 cm to 5 m. One line is a slip. Which?
Noor's 10 cm wall redrawn from 1 cm to 4 m at 1 cm to 5 m
- 110 cm at 1 cm to 4 m, new scale 1 cm to 5 m
- 210 × 4 = 40 m× 4
- 340 × 5 = 200 cm× 5
- AnswerThe new plan length is 200 cm
Problem 9 of 15
Noor's line 3 should turn the real 40 m into a length on the new plan, where 1 cm stands for 5 m. What does it give, in cm?
cm
Problem 10 of 15
Ravi compared two scales: a plan at 1 cm to 3 m, redrawn at 1 cm to 9 m. One line is a slip. Which?
Ravi's comparison of 1 cm to 3 m with 1 cm to 9 m
- 1Old scale 1 cm to 3 m, new scale 1 cm to 9 m
- 29 ÷ 3 = 3, so every length on the drawing is multiplied by 3
- 3A 6 cm wall becomes 6 × 3
- AnswerThe wall is 18 cm on the new plan
Problem 11 of 15
A wall is 6 cm on a plan at 1 cm to 3 m. The plan is redrawn at 1 cm to 9 m. How long is the wall on the new plan, in cm?
cm
Pick the Move
Choose the move before you make it.
Problem 12 of 15
A plan at 1 cm to 8 m is being redrawn at 1 cm to 2 m. A wall is 4 cm on the old plan. Which move finds its length on the new plan?
Problem 13 of 15
The same building is drawn twice: once at 1 cm to 10 m and once at 1 cm to 20 m. Which drawing is smaller, and why?
Same Building?
Decide without drawing anything. Ask what one centimeter stands for each time.
Problem 14 of 15
A wall is 12 cm long on a plan at 1 cm to 2 m. Select every new scale that would draw the same wall shorter than 12 cm.
Problem 15 of 15
A plan is redrawn at a new scale. Select every statement that is true.
Step 7: Quick Check
Show what you know.
Question 1 of 1
9 cm at 1 cm to 2 m, redrawn at 1 cm to 3 m. New length in cm?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A wall is 8 cm long on a plan at 1 cm to 6 m. On a second plan of the same building that wall is drawn 16 cm long. On the second plan, how many meters does one centimeter stand for?
meters
Stretch 2 of 2
A hall is 4 cm by 3 cm on a plan at 1 cm to 5 m. The plan is redrawn at 1 cm to 2 m. How many square centimeters does the hall cover on the new plan?
square cm
What You Learned
- To change scale, go through the real length.
- Old drawing to real, then real to new drawing.
- A scale covering more ground per centimeter makes a smaller drawing.
For the grown-up
The misconception this lesson targets is that a bigger number in the scale makes a bigger drawing. A child holding it multiplies by the new scale number instead of dividing by it, and ends up with a plan longer than the building it shows. A second slip rides alongside it: reading "1 cm to 8 m" as four times as much drawing rather than four times as much ground, so the single factor between two scales gets used upside down. The cure for both is the same sentence, said out loud before each move — plan up to the ground multiplies, ground down to a plan divides — followed by a glance at whether the answer came out the size it should be.
Changing the scale of a drawing — To redraw at a new scale, find the ground measurements from the original drawing, then apply the new scale to each. Going through the real dimensions avoids compounding the two scales incorrectly.
Or work out a single factor — Going from 1 : 100 to 1 : 50 doubles every length on the drawing, because the new scale shows things at twice the size. One factor applied to every measurement is quicker, provided you get the direction right.
Angles never change — Rescaling multiplies lengths and leaves every angle exactly as it was. If an angle changes, the new drawing is not a scale copy. This is the same defining property from the first lesson of the chapter, now used as a check.
The area changes by the square — Doubling every length on a redrawn plan quadruples the area of the drawing, even though the building it represents is unchanged. Keeping clear about which area you mean — drawing or ground — prevents a confusing kind of error.
Find the factor between the scales — A drawing at 1 : 100 is to be redrawn at 1 : 50. The new scale shows things twice as large. So every length on the drawing doubles. A 3 cm wall becomes 6 cm.