1.04a - Undoing calculations

1.04a - Undoing calculations

If you know a calculation's answer, can you recover its starting number? You will reverse addition, subtraction, multiplication, division and simple powers, then use those relationships to check answers, find missing digits and simplify arithmetic.

Undoing addition and subtraction

I think of a number, multiply it by four, then add seven. The result is 3535. What was my number? Keep the order of the instructions in mind: we will undo this machine after learning the inverse pairs.

Start with one operation. Adding 1717 and then subtracting 1717 restores any starting number. For example, −25+17=−8-25+17=-8, and −8−17=−25-8-17=-25. The moves cancel even when the values are negative.

Inverse operations

Inverse operations undo one another. Addition and subtraction of the same number are inverse operations: to undo “add aa”, subtract aa; to undo “subtract aa”, add aa.

A missing box is a number, not an instruction to ignore a term. First identify what happened to the box; then reverse it.

Worked example 1: undoing an addition

Question. Write the missing number: □+17=−8\square+17=-8.

Plan. Subtract seventeen from the result to undo adding seventeen.

□=−8−17undo the addition=−25move seventeen left\begin{aligned} \square&=-8-17 &&\text{undo the addition}\\ &=-25 &&\text{move seventeen left} \end{aligned}

Answer. −25-25.

Check. −25+17=−8-25+17=-8, so it makes the original statement true.

Change only the operation on the box, from addition to subtraction. Its inverse changes too.

Worked example 2: undoing a subtraction

Question. Write the missing number: □−17=−8\square-17=-8.

Plan. Add seventeen to undo subtracting seventeen.

□=−8+17undo the subtraction=9move seventeen right\begin{aligned} \square&=-8+17 &&\text{undo the subtraction}\\ &=9 &&\text{move seventeen right} \end{aligned}

Answer. 99.

Check. 9−17=−89-17=-8.

A different position of the box needs a different reading. In 20−□=720-\square=7, the box is the amount removed, so it is 20−7=1320-7=13; it is not 7+207+20. Check the original statement rather than memorising a box-position rule.

Watch out

To undo subtracting a negative number, add that negative number. For example, undoing “subtract −3-3” means adding −3-3, not adding 33.

Read the operation before choosing its inverse.

Undoing multiplication and division

Multiplication by a nonzero number can be undone by division by that same number.

For example, 7×(−6)=−427\times(-6)=-42, so (−42)÷(−6)=7(-42)\div(-6)=7. Conversely, division by −6-6 can be undone by multiplying by −6-6. Negative numbers do not change which operations are inverse.

Key idea

To undo multiplication by aa, divide by aa; to undo division by aa, multiply by aa, where a≠0a\ne0. Multiplication by zero loses the starting value: every number times zero is zero, so there is no unique way to undo it.

The number being divided matters. In 24÷□=624\div\square=6, ask “six groups of what make twenty-four?” The missing divisor is 44. Multiplying 2424 by 66 would answer a different question.

Worked example 3: undoing a multiplication

Question. Write the missing number: □×(−6)=42\square\times(-6)=42.

Plan. Divide the result by negative six.

□=42÷(−6)undo multiplication=−7different signs\begin{aligned} \square&=42\div(-6) &&\text{undo multiplication}\\ &=-7 &&\text{different signs} \end{aligned}

Answer. −7-7.

Check. (−7)×(−6)=42(-7)\times(-6)=42.

Change only the operation applied to the box. Division now needs multiplication to undo it.

Worked example 4: undoing a division

Question. Write the missing number: □÷(−6)=42\square\div(-6)=42.

Plan. Multiply the result by negative six.

□=42×(−6)undo division=−252different signs\begin{aligned} \square&=42\times(-6) &&\text{undo division}\\ &=-252 &&\text{different signs} \end{aligned}

Answer. −252-252.

Check. (−252)÷(−6)=42(-252)\div(-6)=42, using equal signs.

Return to the opening machine. Its last instruction was “add seven”, so undo that first. Then undo multiplication by four:

35−7=28,28÷4=735-7=28,\qquad28\div4=7

The starting number was 77. Check forwards: 7×4+7=357\times4+7=35. Reversing a chain means reversing both the operations and their order, just as you take your shoes off before your socks.

Watch out

Starting with 35÷435\div4 would undo the first instruction before removing the last one. For the opening machine, subtract seven first, then divide by four.

Use the inverse of the operation shown.

Undoing powers with roots

A square of side length 88 has area 8×8=648\times8=64, so recovering its side length means finding the nonnegative number whose square is 6464.

We write 82=648^2=64. The raised two means multiply two equal factors, and 64=8\sqrt{64}=8 reverses that squaring for a nonnegative starting number. Similarly, 43=4×4×4=644^3=4\times4\times4=64, and 643=4\sqrt[3]{64}=4 reverses cubing.

Square root and cube root

For a≥0a\ge0, a\sqrt a is the principal square root: the nonnegative number whose square is aa. The cube root a3\sqrt[3]a is the number whose cube is aa; negative numbers have negative real cube roots.

There is an important limit to “undoing”. Both 88 and −8-8 square to 6464. Squaring loses the sign, so 64\sqrt{64} returns 88, not two answers. If the task asks for every number whose square is 6464, give both 88 and −8-8. A length must be nonnegative.

Worked example 5: reversing a square

Question. A positive number is squared to give 6464. Find the number.

Plan. Take the principal square root, because the starting number is positive.

8×8=64identify equal factors64=8undo squaring on positives\begin{aligned} 8\times8&=64 &&\text{identify equal factors}\\ \sqrt{64}&=8 &&\text{undo squaring on positives} \end{aligned}

Answer. 88.

Check. Squaring the answer gives 82=648^2=64.

Keep the result 6464 and change only the power from square to cube. The inverse must change with it.

Worked example 6: reversing a cube

Question. A positive number is cubed to give 6464. Find the number.

Plan. Find three equal factors rather than two.

4×4×4=64three equal factors643=4undo cubing\begin{aligned} 4\times4\times4&=64 &&\text{three equal factors}\\ \sqrt[3]{64}&=4 &&\text{undo cubing} \end{aligned}

Answer. 44.

Check. 43=16×4=644^3=16\times4=64.

Unlike squaring, cubing keeps the sign: (−3)3=−27(-3)^3=-27, so −273=−3\sqrt[3]{-27}=-3. Zero has both square root and cube root zero. There is no real square root of a negative number, because a real number squared is never negative.

Watch out

64=8\sqrt{64}=8, not 3232: undoing a square means finding equal factors, not dividing by two. Likewise, undoing a cube is not division by three.

Use the correct number of equal factors.

Missing numbers and efficient arithmetic

Inverse operations can simplify a calculation as well as recover a missing value.

For 374−198374-198, subtracting 200200 is convenient, but removes two too many. Add those two back:

374−198=374−200+2=176374-198=374-200+2=176

This is compensation: adjust an awkward number and undo the adjustment. Regrouping can help too: 37+68+63=(37+63)+68=16837+68+63=(37+63)+68=168. With multiplication, double one factor and halve the other to keep the product unchanged: 25×16=50×8=100×4=40025\times16=50\times8=100\times4=400. Multiplying one factor by two and dividing the other by two cancel. Do not double both.

A missing digit problem adds a place-value restriction. In 4□4\square, the box is the ones digit of a two-digit number, so 4□4\square must be one of 4040 to 4949.

Worked example 7: a missing digit with compensation

Question. Find the digit in 4□+27=754\square+27=75.

Plan. Undo the addition to recover the whole two-digit number, then identify its ones digit.

4□=75−27undo adding 27=75−30+3compensate for extra 3=48recover the whole number□=8read its ones digit\begin{aligned} 4\square&=75-27 &&\text{undo adding 27}\\ &=75-30+3 &&\text{compensate for extra 3}\\ &=48 &&\text{recover the whole number}\\ \square&=8 &&\text{read its ones digit} \end{aligned}

Answer. The digit is 88.

Check. 48+27=48+20+7=7548+27=48+20+7=75.

Keep the total and change only the addend from 2727 to 2828. The missing number must decrease by one to balance the change.

Worked example 8: balancing a one-unit change

Question. Find the digit in 4□+28=754\square+28=75.

Plan. Subtract twenty-eight, then check that the result still begins with four.

4□=75−28undo adding 28=75−30+2compensate for extra 2=47recover the whole number□=7read its ones digit\begin{aligned} 4\square&=75-28 &&\text{undo adding 28}\\ &=75-30+2 &&\text{compensate for extra 2}\\ &=47 &&\text{recover the whole number}\\ \square&=7 &&\text{read its ones digit} \end{aligned}

Answer. The digit is 77.

Check. 47+28=47+20+8=7547+28=47+20+8=75. The recovered number falls by one, as predicted.

A candidate must satisfy both the calculation and the digit position. If undoing the addition gave 5858, no digit could make 4□4\square equal 5858.

Watch out

In 4□+28=754\square+28=75, the missing digit is 77, not 4747. The calculation first recovers the whole number; the question asks for just one digit.

Use a fresh addend and verify the completed number.

Quick Recap

  • Addition and subtraction undo each other; multiplication and division by the same nonzero number undo each other.
  • Reverse a chain by undoing its last operation first.
  • A principal square root is nonnegative; cubing can be reversed for negative numbers too.
  • Compensation changes a calculation and then cancels the change. A missing digit must also fit its place value.

The opening number was 77: 35−7=2835-7=28, then 28÷4=728\div4=7.