doing math

courses › Mental Arithmetic

Grouping & estimating

level 11 course

Add lists smartly and estimate before you calculate.

Best in your head · no pen, no calculator why?

Learn first (about 3 minutes)

Opens at level 11.

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Builds on: Two-digit add & subtract (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Two habits make long calculations safer.

Group before you add. In a list, look for two numbers whose endings make a round hundred, like 275 and 125, and add those first. The order never changes the total, so pick the order that carries least.

Estimate. Round the numbers and calculate with the rounded ones. An estimate is not the exact answer, so when a question says how to round, give the rounded result. Estimates also catch the most common slip: an answer 10 times too big or too small.

Techniques

Pair the round hundreds

Adding a list of three-digit numbers.

  1. Look at the last two digits of each number.
  2. Find two whose endings make 100, like 75 and 25, or 40 and 60.
  3. Add that pair first, then add the rest.
worked example

Example: 340 + 215 + 160

  1. 40 and 60 make 100, so pair 340 and 160.
  2. 340 + 160 = 500.
  3. 500 + 215 = 715.

Answer: 715

Round as the question says

A question asks for an estimate and says how to round.

  1. Round each number the way the question says: to the nearest ten, or the nearest hundred.
  2. For the nearest ten, look at the ones digit: 5 or more rounds up, less rounds down.
  3. If the question offers a friendlier pair of numbers, use that pair.
  4. Calculate with the rounded numbers and give that result, not the exact one.
worked example

Example: Estimate 38 × 62 by rounding each factor to the nearest ten.

  1. 38 rounds to 40, and 62 rounds to 60.
  2. 40 × 60 = 2,400.

Answer: 2,400

Count the digits

Choosing the right size for a product or a division.

  1. For a product, round each number to its first digit, multiply those digits, then add the zeros.
  2. For a division, multiply the divisor by 10, 100 and 1,000 to see which two the number falls between.
  3. Pick the choice that fits. The others are off by about 10 times or more.
worked example

Example: Which is closest to 63 × 288: about 1,800, about 18,000 or about 180,000?

  1. Round to first digits: 60 × 300.
  2. 6 × 3 = 18, then three zeros: 18,000.

Answer: 18,000

Tips by skill

  • TipGroup into hundreds: Find the two numbers whose last two digits make 100. Add them first, then the third.
  • TipEstimate by rounding: Round exactly as the question says, then calculate with the rounded numbers, not the originals.
  • TipRight size?: Round to first digits and count the zeros. For a division, check what the divisor times 10, 100 or 1,000 gives.

Watch out for

  • Giving the exact answer when the question asks for an estimate.
  • Dropping the hundred carried from a pair. 275 + 125 is 400, not 300.
  • Adding or losing a zero. 40 × 400 is 16,000: 4 × 4 = 16, then three zeros.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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