The 100-grid: chains of changes and points in squares

Lesson 9 of 1215 min5 practice questionsFree

A grid of 10 by 10 has 100 small squares, so each square is 1%. Make it 30% longer and 20% wider and you add a strip of 30 squares, a strip of 20 squares and a corner of 6 squares where the strips meet: the area grows 56%, not 50%. The same grid shows percentage points too: squares added out of 100.

Guess first

Guess first: a rectangular garden is made 30% longer and 20% wider. By what percent does its area grow?

A 50% increase

B 56% increase

C 44% increase

D 6% increase

Show answer

Correct answer: B

Picture the old garden as 100 squares (10 by 10). 30% longer adds a strip of 3 × 10 = 30 squares; 20% wider adds a strip of 10 × 2 = 20 squares; where the strips meet, a corner of 3 × 2 = 6 squares is new too. 100 + 30 + 20 + 6 = 156 squares: up 56%. Check: 1.3 × 1.2 = 1.56.

Option A: 50% adds the two percents and forgets the corner, the 6 squares where the strips meet.

Option C: 44% takes the corner away instead of adding it (50 − 6 = 44). Both changes are rises, so the corner is extra area.

Option D: 6% is only the corner; the two strips are missing.

Hint: Draw the 10-by-10 grid and add both strips. What fills the corner?

If you said 56%, you saw the corner. This lesson is about that corner: it is the reason why changes in a row multiply instead of adding.

An everyday picture

The grid works like the tiles of a floor: 100 equal tiles in 10 rows of 10.

The old floor of 100 tiles 100%

each tile is 1% of the floor

Make the room 30% longer 30 more tiles

3 more tiles in each of the 10 rows

Then make it 20% wider 26 more tiles

two new rows of 13: 10 tiles each for the old length and 3 each for the new length

The 6 extra tiles at the end of the two new rows are the change on the change: the new width also runs along the new length.

See it: two changes on one grid

Length up 30%, width up 20%. Dark squares are the old 100, open squares are the two strips, crossed squares are the corner:

\def\arraystretch{0.8}\def\Bq{\blacksquare}\def\Wq{\square}\def\Cq{{\boxtimes}}\begin{array}{l} \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Cq\Cq\Cq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Cq\Cq\Cq \end{array}

Count them: 100 + 30 + 20 + 6 = 156 squares (100 old, 30 in the length strip, 20 in the width strip, 6 in the corner). The area is up 156 − 100 = 56 squares, that is 56%. The multipliers of lesson 3 do the same count in one step: 1.3 × 1.2 = 1.56.

Two changes	Strips (squares)	Corner (squares)	Area
up 30%, up 20%	30 + 20 = 50	30 × 20 ÷ 100 = 6, added	up 56%
up 40%, up 10%	40 + 10 = 50	40 × 10 ÷ 100 = 4, added	up 54%
up 50%, down 20%	50 − 20 = 30	50 × 20 ÷ 100 = 10, taken away	up 20%
down 20%, down 20%	20 + 20 = 40 taken away	20 × 20 ÷ 100 = 4, given back	down 36%

flowchart TD
  A["100 squares"] -->|"+30: length strip"| B["130"]
  B -->|"+20: width strip"| C["150"]
  C -->|"+6: the corner"| D["156: up 56%"]

The same grid shows percentage points. In a village, 32 of every 100 homes had a solar lamp; a year later, 44 did. Dark squares are last year's 32 homes, squares with a plus are the 12 new ones, open squares are homes without a lamp:

\def\arraystretch{0.8}\def\Bq{\blacksquare}\def\Wq{\square}\def\Pq{{\boxplus}}\begin{array}{l} \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq \\ \Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq\Bq \\ \Bq\Bq\Pq\Pq\Pq\Pq\Pq\Pq\Pq\Pq \\ \Pq\Pq\Pq\Pq\Wq\Wq\Wq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq \\ \Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq\Wq \end{array}

The share went up by 12 squares out of 100: 12 percentage points. As a percent of last year's 32 squares, the rise is 12 ÷ 32 × 100 = 37.5%.

Try it. Draw the grid for a plot whose length grows 10% and whose width grows 30%. How many squares are in each strip, and how many in the corner?

Find the rule

Both sides of a square photo are enlarged by 30%. By what percent does its area grow?

A 60% increase

B 69% increase

C 9% increase

D 51% increase

Show answer

Correct answer: B

Strips: 30 + 30 = 60 squares. Corner: 30 × 30 ÷ 100 = 9 squares. 60 + 9 = 69, so the area grows 69%. Check: 1.3 × 1.3 = 1.69.

Option A: 60% adds the two strips and forgets the 9-square corner.

Option C: 9% is only the corner; the strips are missing.

Option D: 51% takes the corner away (60 − 9 = 51). Both sides grew, so the corner is extra area.

Hint: Each strip is 30 squares. How big is the corner?

The price of a notebook rises 20% and a student buys 10% fewer notebooks. By what percent does the student's spending on notebooks change?

A 10% increase

B 8% increase

C 12% increase

D 2% decrease

Show answer

Correct answer: B

Spending = price × number. On the grid, the price strip adds 20 squares and the number strip takes 10 away; the corner, 20 × 10 ÷ 100 = 2, is taken away too: the price strip counted the extra 20% on the notebooks that were not bought. 100 + 20 − 10 − 2 = 108: up 8%. Check: 1.2 × 0.9 = 1.08.

Option A: 10% adds the percents (20 − 10 = 10) and forgets the corner.

Option C: 12% adds the corner (10 + 2 = 12). One change is a fall, so the corner is taken away.

Option D: A 2% decrease is only the corner, with its sign; the strips are missing.

Hint: One strip grows and one shrinks. Is the corner added or taken away?

On a 100-square grid of a school's budget, books had 16 squares last year and 20 squares this year. Which statement is right?

A Books went up by 4%

B Books went up by 4 percentage points, which is a 25% rise

C Books went up by 25 percentage points

D Books went up by 20%

Show answer

Correct answer: B

20 − 16 = 4 squares out of 100: 4 percentage points. As a percent of last year's 16 squares: 4 ÷ 16 × 100 = 25%.

Option A: 4 is the change in points; as a percent of 16 squares it is 25%.

Option C: 25 is the percent rise, not the points: only 4 squares were added.

Option D: 20% divides the 4 squares by this year's 20 (4 ÷ 20 × 100 = 20%); the rise is measured on last year's 16.

Hint: Count the squares added, then divide by the squares before.

Name the rule

Two changes on one product (area = length × width, spending = price × number, two years of growth): the change is a + b + a × b ÷ 100, with each fall written as a minus.
The corner, a × b ÷ 100, is the change on the change. It is added when both changes rise, taken away when one rises and one falls, and given back when both fall.
Percentage points = squares added out of 100. Percent change = squares added ÷ squares before × 100.
The multipliers do the whole count in one step: (1 + a ÷ 100) × (1 + b ÷ 100).

Why is there a corner at all?

Because the second change acts on the new size, not on the old one. The room grows 20% in width along its NEW length, which is 30% longer; the bit of the new rows that runs along the new 30% is the corner.

Worked, half-done, solo

Question

A shop raises the price of a tea packet by 25% and then sells 10% fewer packets. By what percent do its takings from tea change?

Solution

Write the two changes
Takings = price × packets sold: up 25% and down 10%.
Add the strips
25 − 10 = 15 squares.
Find the corner
25 × 10 ÷ 100 = 2.5 squares, taken away because one change is a fall.
Put it together
15 − 2.5 = 12.5, so the takings rise 12.5%. Check: 1.25 × 0.9 = 1.125.

Answer: 12.5% increase

Watch out: 15% adds the percents and forgets the corner.

Now finish one that is half done.

A rectangle's length grows 30% and its width shrinks 20%. Strips: 30 − 20 = 10 squares. Corner: 30 × 20 ÷ 100 = 6 squares. What happens to the area?

A 4% increase

B 10% increase

C 16% increase

D 6% decrease

Show answer

Correct answer: A

One change is a fall, so the corner is taken away: 10 − 6 = 4. The area grows 4%. Check: 1.3 × 0.8 = 1.04.

Option B: 10% stops at the strips and forgets the corner.

Option C: 16% adds the corner (10 + 6 = 16) instead of taking it away.

Option D: A 6% decrease is only the corner; the strips are missing.

Hint: Is the corner added or taken away when one side shrinks?

And one on your own.

A town's population grows 5% in one year and 20% in the next. By what percent has it grown over the two years?

A 25%

B 26%

C 12.5%

D 1%

Show answer

Correct answer: B

Strips: 5 + 20 = 25 squares. Corner: 5 × 20 ÷ 100 = 1 square. 25 + 1 = 26, so the population grew 26%. Check: 1.05 × 1.2 = 1.26.

Option A: 25% adds the two percents and forgets the 1-square corner.

Option C: 12.5% is the average of the two changes ((5 + 20) ÷ 2 = 12.5); the changes act one after the other, they are not averaged.

Option D: 1% is only the corner.

Hint: The second year's 20% is of a population that is already 5% bigger.

The traps

Five ways to lose this mark

Forgot the corner: two changes add their strips AND a corner; 30% and 20% make 56%, not 50%.
Corner with the wrong sign: when one change rises and the other falls, the corner is taken away.
Only the corner: a × b ÷ 100 is the extra bit, not the whole change.
Points called percent: squares added out of 100 are percentage points; as a percent of the squares before, the rise is bigger.
Wrong base for the percent: divide the squares added by the squares before, not after.

Ritu says: "The price rises 40% and I buy 25% less, so my spending rises 40 − 25 + 10 = 25%." What is right?

A Up 5%: the corner, 40 × 25 ÷ 100 = 10, is taken away

B Ritu is right: up 25%

C Up 15%: there is no corner

D Down 10%: the corner alone

Show answer

Correct answer: A

One change rises and one falls, so the corner is taken away: 40 − 25 − 10 = 5. Spending rises 5%. Check: 1.4 × 0.75 = 1.05.

Option B: 25% adds the corner instead of taking it away.

Option C: 15% adds the percents and forgets the corner.

Option D: A 10% fall is only the corner; the strips are missing.

Hint: One change is a fall. What happens to the corner?

Mixed check

A city's bus fare rises 20%. By what percent must a student cut the number of trips so that the month's spending on fares stays the same?

A 16⅔% fewer

B 20% fewer

C 25% fewer

D 83⅓% fewer

Show answer

Correct answer: A

In blocks of 20%, the old fare is 5 blocks and the new fare 6 blocks. To keep the spending, the trips must take 1 block of the 6 away: 1 ÷ 6 × 100 = 16⅔%. Check: 1.2 × 5/6 = 1.

Option B: 20% fewer undoes 20% of the OLD fare, but the cut is measured on the new 6 blocks.

Option C: 25% fewer uses the rule for undoing a fall (20 ÷ 80 × 100 = 25); after a rise, divide by the 6 blocks of the new fare.

Option D: 83⅓% is the share of trips the student still makes (5 ÷ 6 × 100 = 83⅓), not the cut.

Hint: The fare went from 5 blocks to 6. What share of 6 must go?

From lesson 8:

After a 25% fall, a scooter costs ₹48,000. What did it cost before the fall?

A ₹60,000

B ₹64,000

C ₹36,000

D ₹16,000

Show answer

Correct answer: B

25% = 1/4, so the old price is 4 blocks and the new price is 3 blocks. 1 block = 48,000 ÷ 3 = 16,000, so the old price = 4 × 16,000 = ₹64,000.

Option A: ₹60,000 adds 25% of ₹48,000 back. The 25% was of the old price, which is bigger.

Option C: ₹36,000 takes another 25% off instead of undoing the fall.

Option D: ₹16,000 is one block, the fall itself, not the old price.

Hint: The new price is 3 blocks.

Quick reference

The 100-grid

Two changes on one product: a + b + a × b ÷ 100 up 30% and up 20%: 30 + 20 + 6 = 56, up 56%
The corner: a × b ÷ 100: added, or taken away when one change falls up 20%, down 10%: 20 − 10 − 2 = 8, up 8%
Spending stays the same: the quantity takes 1 block of the new price away fare up 20%: 1 ÷ 6 × 100 = 16⅔% fewer trips
Percentage points: squares added out of 100 16 → 20 squares: up 4 points
Percent change: squares added ÷ squares before × 100 4 ÷ 16 × 100 = 25%

Why it works: the second change acts on the new size, so the two strips meet in a corner, and that corner is the change on the change.

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Start practice: Lesson 9 practice: The 100-grid

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