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quick techniques

Make ten A sum crosses a ten, a hundred or a thousand.
Making ten: 38 + 7A number line. 7 chips: 2 of them take 38 up to 40, and the other 5 carry on to 45. 38 + 7 = 45.3050404075 left+2+53845
  1. Top the bigger number up to the next ten.
  2. Take that much from the other number.
  3. Add what is left.

example 38 + 7

  1. 38 needs 2 to reach 40.
  2. 7 − 2 = 5 left.
  3. 40 + 5 = 45.

Answer: 45

used in 3 courses

Small facts · Gaps & place value · Two-digit add & subtract

Doubles and near doubles The two numbers are the same or close together.
Near doubles: 25 + 26Two bars, 25 and 26 long. The shared 25 of each doubles to 50; the 1 left over makes 51.2526+125 + 25 = 5050 + 1 = 51
  1. Double the smaller one.
  2. Add the difference.

example 25 + 26

  1. 25 + 25 = 50.
  2. One more: 50 + 1 = 51.

Answer: 51

used in 2 courses

Small facts · Two-digit add & subtract

Count up to subtract Subtracting, making change, or finding a gap.
Counting up from 64 to 100A number line. A jump of 6 from 64 to 70, then A jump of 30 from 70 to 100. Together: 36.608090+6+3064701006 + 30 = 36
  1. Start at the smaller number.
  2. Jump to the next round number, unless the bigger number comes first.
  3. Jump on to the bigger number.
  4. Add the jumps.

example 100 − 64

  1. 64 up to 70 is 6.
  2. 70 up to 100 is 30.
  3. 6 + 30 = 36.

Answer: 36

used in 4 courses

Small facts · Gaps & place value · Two-digit add & subtract · Money & prices

Round, then adjust A number sits just below a round one, like 29, 98 or $1.99.
Round, then adjust: 83 − 29A number line. From 83, take away 30 to reach 53, then add back the 1 the rounding changed: 54.607090−30+1835353 + 1 = 54
  1. Use the round number instead.
  2. Work out the answer.
  3. Put back what the rounding changed. In a product, multiply the difference by the other number: 29 × 4 = 120 − 4.

example 83 − 29

  1. 83 − 30 = 53.
  2. You took away 1 too many, so add it back: 53 + 1 = 54.

Answer: 54

used in 3 courses

Two-digit add & subtract · Bigger multiplication · Money & prices

Split and combine Multiplying a two-digit number by a one-digit number.
Split and combine: 7 × 46A 7 by 46 rectangle, cut into 40 and 6: 280 and 42, which add to 322.7467 × 40 = 2807 × 6 = 4240280 + 42 = 322
  1. Split the bigger number into tens and ones.
  2. Multiply each part.
  3. Add the parts.

example 7 × 46

  1. 7 × 40 = 280.
  2. 7 × 6 = 42.
  3. 280 + 42 = 322.

Answer: 322

used in 2 courses

Bigger multiplication · Division

Double and halve One number is even and the other ends in 5.
Double and halve: 16 × 25A 16 by 25 rectangle cut in half and laid end to end: 8 by 50, then 4 by 100. The area stays 400.162585041004 × 100 = 400
  1. Halve the even number.
  2. Double the other one.
  3. Repeat until the product is easy.

example 16 × 25

  1. 16 × 25 = 8 × 50.
  2. 8 × 50 = 4 × 100.
  3. 4 × 100 = 400.

Answer: 400

used in 2 courses

Times tables · Bigger multiplication

Shift for 10, 100 and 1,000 Multiplying or dividing by 10, 100 or 1,000.
3.2 × 100 on a place-value chartThe digits of 3.2 slide two places left past the fixed decimal point and read 320.hundredstensonestenths320× 1003.2 × 100 = 320
  1. Times 10 moves every digit one place to the left: 45 becomes 450.
  2. Dividing by 10 moves every digit one place to the right: 45 becomes 4.5.
  3. Count the zeros to know how many places to move.

example 3.2 × 100

  1. 100 has two zeros, so move two places left.
  2. 3.2 becomes 320.

Answer: 320

used in 3 courses

Gaps & place value · Division · Decimals

Build from 10%, 5% and 1% Finding any percent of an amount.
15% of $80, built from easy piecesA bar for $80 cut into tenths. 10% is $8; 5% is $4. Together, 15% is $12.10% is $85% is $415% is $12$80 in all
  1. 10% is the amount divided by 10.
  2. 5% is half of 10%.
  3. 1% is the amount divided by 100.
  4. Add the pieces you need: 15% = 10% + 5%.

example 15% of $80

  1. 10% of $80 is $8.
  2. 5% is half of that: $4.
  3. $8 + $4 = $12.

Answer: $12

used in 3 courses

Percents · First Business Story · Percentages in Business

Know the equivalents A percent is a simple fraction, or a fraction is a simple percent.
25% of 360 as 1/4A bar for 360 cut into 4 equal parts. 25% is 1/4, one part: 90.3601/4901/4901/4901/490360 ÷ 4 = 90
  1. Learn the anchors: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%.
  2. Eighths are half of quarters: 1/8 = 12.5%.
  3. 1/3 and 2/3 never end: 1/3 ≈ 33.3% and 2/3 ≈ 66.7%.

example 25% of 360

  1. 25% is 1/4.
  2. 360 ÷ 4 = 90.

Answer: 90

used in 3 courses

Fraction basics · Percents · Division

Estimate first Before a long calculation, and to check an answer.
Estimating 412 × 38412 × 38 rounds to 400 × 40 = 16,000. On a line from 0, the exact 15,656 lands right beside it.412 × 38≈ 400 × 40010,00020,000estimate 16,000exact 15,656
  1. Round each number to its first digit.
  2. Do the easy calculation.
  3. Your exact answer should land near it. If it is 10 times off, a zero or a decimal point slipped.

example Round each number to its first digit, then multiply: 412 × 38

  1. 412 rounds to 400, and 38 rounds to 40.
  2. 400 × 40 = 16,000.
  3. The exact answer, 15,656, is close.

Answer: 16,000

used in 3 courses

Grouping & estimating · Orders of magnitude · Fermi estimates

Work backward from a percent You know the amount after a percent change and want the amount before it.
Working back from $96 after 20% offA bar for the original price, cut into tenths. $96 fills 80% of it, $12 a slice, so the whole bar is $120.original = ?original $120$12$12$12$12$12$12$12$12$12$12$96 is 80%20% is $24$120 − $24 = $96
  1. Say what share of the original you have: 20% off leaves 80%.
  2. Divide by that share, written as a decimal.
  3. Never add the percent back on. That uses the wrong base.

example After 20% off, a coat costs $96. What was the original price?

  1. $96 is 80% of the original.
  2. $96 ÷ 0.8 = $120.
  3. Check: 20% of $120 is $24, and $120 − $24 = $96.

Answer: $120

used in 4 courses

What percent? Work backward · Percentages in Business · Markup Is Not Margin · Invoices & Sales Tax

Pretend it is 100 A percent question gives no amounts at all.
Starting from 100Bars from a start of 100: up 10% to 110, then down 10% to 99. The end is 1 below 100: −1%.9010011010011099up 10%down 10%start−1%
  1. Start with 100, or $100.
  2. Apply each change in order.
  3. Read the answer off your numbers: the gap from 100 is the percent change. A margin is profit ÷ price.

example A price goes up 10%, then down 10%. What is the overall change, in percent?

  1. 100 up 10% is 110.
  2. 110 down 10% is 99.
  3. 99 is 1 below 100: a 1% drop.

Answer: −1%

used in 3 courses

Percentages in Business · Markup Is Not Margin · Money & prices

Keep the balance Solving any equation.
Balancing 2x + 9 = 25A level balance: 2 x boxes and 9 dots on the left, 25 dots on the right. Take 9 from both sides: 2x = 16. Share 16 into 2 equal groups: x = 8.x8x888− 9− 92x + 9 = 252x = 16x = 82 × 8 + 9 = 25left sideright side
  1. Whatever you do to one side, do to the other.
  2. Undo in reverse order: in 2x + 9, the 9 came last, so take it away first.
  3. Check by putting the answer back in.

example Solve for x: 2x + 9 = 25

  1. Take 9 from both sides: 2x = 16.
  2. Divide both sides by 2: x = 8.
  3. Check: 2 × 8 + 9 = 25.

Answer: 8

used in 3 courses

Expressions & one-step equations · Linear equations · Solve for the Unknown

Free tools: calculators for markup, margin, break-even and percent change, and a mental math speed drill. No sign-up.

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