Equations & formulas — cards that show the working, not just the answer

"Choose 2 of 5", top left, is a permutations & combinations card. It has n set to 5 and r to 2, and shows the working in order: ₅C₂ = 5! ÷ (2! × 3!) = (5 × 4) ÷ (2 × 1) = 10.

"Choices that include A", bottom left, is the same card. A's partner is chosen from the other 4 people, so there are ₄C₁ = 4 ways.

The fraction card on the right is connected first to the bottom-left card, then to the top-left one. The first connection is the numerator and the second the denominator, so the card shows 4/10 reduced to 2/5 in large type. Below it are the decimal, 0.4, and the percentage, 40%.

n and r are typed into the combination cards. For 6 people, n becomes 6 on the top-left card and 5 on the bottom-left one. That is 5 ways out of 15, and the fraction card shows 1/3 (33.33%). Setting n back to 5 and 4 brings back 2/5.

Equation

This card solves a linear equation in x, writing out each step of moving terms from one side to the other. The equation is typed into the box on the card. It cannot use values from other cards; drawing a connection to it brings up a notice that the card cannot receive values.

"Number of apples", on the left of Board 2, is the equation for buying some apples at 120 each plus a 200 basket, for 800 in all. With 120x + 200 = 800 typed in, "Subtract 200 from both sides" gives 120x = 600, and "Divide both sides by 120" gives the answer, x = 5.

• Writing the equation — The multiplication sign can be left out, as in 120x. Division is written with /, as in x/3, and brackets are expanded first. Decimals are turned into fractions, and an answer that does not come out whole is given as a fraction, such as x = 2/3 for 3x = 2. An equation with x in a denominator, such as 1/x = 2, starts with the step "Multiply both sides by x to clear the denominator (x ≠ 0)".
• Step by step, and checking — "Step by step" shows only the first step; ◀ and ▶ move one step forward or back, and "Show all" brings back every step. "Check it" puts the answer back into the equation and shows both sides. On the left of Board 2, both are 800.
• No single answer — When x cancels out and the statement is false, as in x + 1 = x + 2, the card says there is no solution. When it is always true, as in 2(x + 1) = 2x + 2, it says it is true for every number.
• Equations it cannot read — With both x and y, the card suggests "Two equations"; with x², it suggests "Quadratic equation". The unknown is written as x; with a letter such as a, the card asks for it to be rewritten as x.
• Passing the answer on — The answer is not passed to other cards. The circle for drawing a connection does not appear when the card is selected.

"System of equations", "Quadratic equation" and "Inequality" in the "Add a card" list give this same equation card. The menu at the top of the card switches between "One equation", "Two equations", "Quadratic equation" and "Inequality" at any time. The text in the box stays as it is when the type changes.

System of equations

This is the equation card set to "Two equations". It has two boxes and solves a pair of simultaneous equations in x and y. As with the equation card, the answer is not passed to other cards.

"Number of stamps", on the right of Board 2, is for buying 10 stamps in all, some at 50 and some at 80, for 620. It has x + y = 10 and 50x + 80y = 620. The card first multiplies ① by 50 and ② by 1 so that both x coefficients are 50, then uses ①−② to eliminate x, which leaves −30y = −120, so y = 4. Putting y = 4 into ② gives the answer, x = 6, y = 4.

• Method — The card matches coefficients and adds or subtracts the two equations to remove one unknown (elimination). If one equation is already in a form such as x = 3, it is substituted straight into the other. An equation such as y = 2x + 1 is rearranged to −2x + y = 1 first.
• Check it — Shows both equations with the answer put in. On the right of Board 2, ① gives 10 = 10 and ② gives 620 = 620.
• No single answer — For x + y = 1 with x + y = 2, the card says the two equations are parallel lines with no solution. For x + y = 1 with 2x + 2y = 2, it says they are the same line, with infinitely many solutions.
• Number of unknowns — The unknowns are x and y; an equation with z cannot be read. For three unknowns, the System of 3 (matrix) card solves the system.

Quadratic equation

This is the equation card set to "Quadratic equation". x² is typed as x^2. The card factors the equation when it can, and otherwise uses the discriminant and the quadratic formula. The answers are not passed to other cards.

The equation already in a new card, x² − 5x + 6 = 0, becomes (x − 2)(x − 3) = 0 with the step "Factor". After "If a product is 0, one factor is 0", the answer is x = 2, 3. For x² − 2x − 1 = 0, the card works out the discriminant, D = 8, then goes on to the quadratic formula. The answer is x = 1 ± √2, with the approximate values 2.414 and −0.4142.

• Form of the equation — The right side does not have to be 0. x² = 2x + 8 is rearranged to x² − 2x − 8 = 0, and x(x + 3) = 40 to x² + 3x − 40 = 0, before solving.
• When the formula is used — When the coefficient of x² is not 1, the card may use the quadratic formula even for an equation that factors. 2x² + 3x − 2 = 0 goes through D = 25 to x = −2, 1/2.
• One answer, or no real answer — x² − 4x + 4 = 0 is (x − 2)² = 0, with the single answer x = 2. When D is negative, the card says there is no real solution.
• Equations it cannot read — Without x², the card suggests solving it as "One equation". An equation with x³ cannot be solved. The unknown is written as x.
• Check it — Shows the left side with each answer put in. If both are 0, both numbers are answers. For answers with √, the value is approximate, shown as "≈ 0".

The next board finds the width of a flower bed that is 3 m longer than it is wide, with an area of 40 m². x(x + 3) = 40 gives x = −8, 5, and the one that works as a length is 5 m.

Open the quadratic equation board in CalcAnyway ↗

Inequality

This is the equation card set to "Inequality". It solves a linear inequality step by step, in the same way as an equation. The signs are typed as <, >, <= and >=, and the card shows them as ＜ ＞ ≦ ≧. The answer is not passed to other cards.

The inequality already in a new card, 5 − 2x <= 11, is rewritten as −2x + 5 ≦ 11, and subtracting 5 gives −2x ≦ 6. The next step is "Divide both sides by −2 (dividing by a negative flips the sign)", and the answer is x ≧ −3.

• Number line — Under the answer, a number line shows the range of x. A filled dot means the boundary is included (≦, ≧); an open dot means it is not (＜, ＞).
• Check it — Picks one number inside the range and puts it into the inequality to show that it holds. For 3 − x > 1 (answer x < 2), it puts in x = 1 and shows that the left side, 2, is greater than the right side, 1.
• Always or never true — When x cancels out, the card says the inequality is true for every number, or that there is no solution.
• Inequalities it cannot read — Two signs in one line, as in 1 < 2x + 3 < 7, or an x², cannot be read.

The next board works out how many items at 300 each can be bought within 2,000, with 500 for delivery on top. 300x + 500 <= 2000 gives x ≦ 5, so up to 5 items. "Check it" puts in x = 4 and shows that the left side, 1700, is at most the right side, 2000.

Open the inequality board in CalcAnyway ↗

System of 3 (matrix)

This card solves simultaneous equations in three unknowns, x, y and z, in matrix form. It is the "Matrix calc" card with its operation set to "Solve the system"; its other operations are described under Matrix calc.

The 3-by-3 grid on the left takes the coefficients of x, y and z, one row per equation. The column in the middle takes the right side of each equation. Once every cell is filled, the column on the right shows the answers, x, y and z from the top. A new card starts with empty cells and a message asking for every cell to be filled.

The next board has x + y + z = 600, 2x + y + 3z = 1300 and x + 3y + 2z = 1300. The answers, from the top, are 100, 200 and 300.

Open the three-equation board in CalcAnyway ↗

• Form of the answers — Answers that do not come out whole are given as fractions. Cells accept numbers such as 1/2, 0.5 and −1.
• Fewer unknowns — The − and + buttons next to "Rows" and "Cols" change the size of the grids. With 2 rows and 2 columns, the card solves two equations in two unknowns: for 2x + y = 1 and x + 3y = 2, the answers are 1/5 and 3/5.
• Solve it myself — Empties the answer cells so an answer can be typed in. The card shows how many are right, such as "2/3 correct", and "All correct" when every one is. "Show the answer" brings the answer back.
• No single answer — Whether there is no solution or infinitely many, the card says there is no single answer (no inverse). A cell that is not a number brings up "Some cells are not numbers".
• Working and other cards — Unlike the equation card, this card shows no steps. Its answers cannot be used in other cards' formulas.

Column math

This card writes out addition, subtraction, multiplication or division in columns, as on paper. The menu at the top picks the operation, and the two numbers are typed into the card. The answer is not passed to other cards.

"Long multiplication", on the left of Board 3, is 123 × 45. It writes 123 × 5 = 615 and 123 × 4 = 492, the second shifted one place, and adds them to 5535.

• Carrying and borrowing — In addition, a carried 1 is written small above the column. In subtraction, the digits changed by borrowing are written small above: for 1000 − 1, they are 0, 9, 9 and 10.
• Division — The quotient goes on top, each subtraction below, and the number left at the bottom is the remainder. For 100 ÷ 7, the quotient is 14 and the remainder 2.
• Step by step — Moves one digit or one row at a time, with the calculation for that step shown underneath, such as "7 ＋ 8 ＝ 15" or "10 ÷ 7 = 1 remainder 3".
• Solve it myself — For addition, subtraction and multiplication, the answer row turns into empty boxes, one per digit. With every box filled, the card shows "Correct!" or "Try again".
• Numbers it takes — Whole numbers from 0 up to 999999, with a divisor of at most 999. Decimals and negative numbers bring up "Enter whole numbers of 0 or more", and dividing by 0 brings up "Cannot divide by 0". A subtraction that takes a bigger number from a smaller one brings up a message that it cannot be done.

Prime factors

This card divides a number by primes until it is written as a product of primes. It lays out the division as a ladder: each prime on the left, and the result of dividing written underneath. The result is not passed to other cards.

With "One number", 60 is divided by 2, 2 and 3, leaving 5, so 60 = 2² × 3 × 5. The card also shows the number of divisors, 12; "Show divisors" lists them: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. A prime such as 97 is reported as a prime number.

With "Two numbers", both numbers are divided by the same prime for as long as both can be. "84 and 36", on the right of Board 3, is divided by 2, 2 and 3, leaving 7 and 3. The primes on the left multiply to 12, the GCD (greatest common divisor); 12 times 7 times 3 is 252, the LCM (least common multiple).

• Step by step — Moves one division at a time, such as "60 ÷ 2 ＝ 30" or "Divide 84 and 36 by 2".
• Numbers it takes — Whole numbers from 1 to 999999; anything else brings up "Enter a whole number from 1 to 999999". For 1, the card says that 1 is neither prime nor composite.
• Using the result in a formula — A calc card's gcd and lcm give the GCD and LCM for use in other calculations (Formulas and functions).

Fraction

This card turns two numbers into a fraction and shows it reduced, as a mixed number, as a decimal and as a picture. The numerator and denominator are either typed into the two boxes on the card or taken from two connected cards. When connecting, the first card is the numerator and the second the denominator.

"Chance that A is chosen" on Board 1 shows 4/10 reduced to 2/5 in large type, with the unreduced 4/10 small above it. Below that come the decimal and the percentage (0.4, 40%), and at the bottom a tape diagram.

• Mixed numbers — When the numerator is larger than the denominator, 7/3 is shown as the mixed number 2 1/3.
• Picture — The menu at the bottom switches the picture between "Tape", "Circle" and "Line".
• Decimals and negative numbers — 0.5/0.25 is treated as 50/25 and reduces to 2. −6/4 is −1 1/2.
• A denominator of 0 — The card shows "÷0" instead of a fraction and passes no value on.
• Passing the value on — The fraction as a decimal is passed to other cards: 0.4 on Board 1.
• Only one connection — The card keeps the fraction typed into its boxes. It uses connected values only when two cards are connected.

Division and fractions uses this card to write the answer to a division as a fraction.

Statistics steps

This card lays out the working for the mean, variance and standard deviation of the values connected to it, as a table and a list of formulas. Values come from list, input, calc and other cards connected by lines; several cards can be connected at once. With a list, the card uses the column the list is totalling.

Board 4 connects a list of five scores (60, 70, 80, 90, 100). In the table, "Deviation" is each score minus the mean, 80, and "Squared" is that number squared. Below the table come the count n (5), the sum Σx (400), the mean (80), the sum of squares (1,000), the variance (200) and the standard deviation (14.1421).

• The value passed on — The left menu at the top picks "Std. deviation", "Variance" or "Mean". The chosen value is shown large at the bottom and passed on to other cards: 14.142136, the standard deviation, on Board 4.
• What the variance divides by — The right menu picks what the variance divides by, "n (population)" or "n−1 (sample)". With "n−1 (sample)" on Board 4, 1,000 is divided by 4: the variance becomes 250 and the standard deviation 15.8114.
• Rows in the table — The table lists the first 12 values, and a line such as "…2 more" counts the rest. The calculation uses every value.
• Too few values — With nothing connected, the card asks for value cards to be connected. With "n−1 (sample)" and a single value, the variance row says it needs 2 or more values.

Mean and median uses this card for the sum and the mean of test scores and of household savings.

Permutations & combinations

This card counts combinations, ways of choosing r out of n (ₙCᵣ), and permutations, ways of choosing and arranging them (ₙPᵣ), showing the working. The menu at the top picks "Combination ₙCᵣ (choose)" or "Permutation ₙPᵣ (arrange)", and n and r are typed into the card. They cannot come from other cards.

Switching "Choose 2 of 5" on Board 1 to permutations gives ₅P₂ = 5! ÷ 3! = 5 × 4 = 20. If the 2 people are also put in order, such as first and second on duty, there are 20 ways. Switching back to combinations brings back 10.

• Choosing and leaving out — For combinations, the card adds the matching pair with the same count, as in "Choosing and leaving out give the same count: ₅C₂ = ₅C₃". ₅C₃ is also worked out with the smaller number, 2, as (5 × 4) ÷ (2 × 1).
• Pascal's triangle — For n up to 12, the card also draws Pascal's triangle. The answer is highlighted, and so are the two numbers above it that add up to it: 4 and 6 above the 10 for ₅C₂ on Board 1.
• Numbers it takes — n is a whole number from 0 to 100. A number outside that brings up "Enter a whole number from 0 to 100", and the card keeps the previous number. When r is larger than n, the card says that r must be at most n.
• Passing the value on — The count is passed to other cards. On Board 1, the fraction card uses the counts from both combination cards.

Articles that use these cards

• Division and fractions — quotient and remainder, fraction, decimal — all three boards use the fraction card, showing the answer to 13 ÷ 4 as a mixed number and a tape diagram.
• Mean and median — how they differ and how to find each — the statistics steps card finds the sum and the mean of eight test scores and of ten households' savings.

Read next — Graphs & functions; Matrices, vectors & complex Describes the cards for function graphs, the normal distribution, the unit circle and number lines, and for matrices, vectors and complex numbers.

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