randRange(0, 1) randRange(2, 3) randRange(2, 9) randRange(2, 9) randRange(2, 9) randRangeExclude(-9, 9, [0, D1, -D1]) randRangeExclude(-9, 9, [0, D2, -D2]) randRangeExclude(-9, 9, [0, D3, -D3]) getGCD(N1, D1) getGCD(N2, D2) getGCD(N3, D3) ORDER ? (N1 / F1) * (D2 / F2) : (N2 / F2) * (D3 / F3) ORDER ? (D1 / F1) * (N2 / F2) : (D2 / F2) * (N3 / F3) (N1 / F1) * (D2 / F2) * (NUMS === 2 ? 1 : (ORDER ? D3 / F3 : N3 / F3)) (D1 / F1) * (N2 / F2) * (NUMS === 2 ? 1 : (ORDER ? N3 / F3 : D3 / F3)) getGCD(NUMERATOR, DENOMINATOR)

`fraction(N1, D1) \div fraction(N2, D2) \left(fraction(N1, D1) \div fraction(N2, D2)\right) \div fraction(N3, D3) fraction(N1, D1) \div \left(fraction(N2, D2) \div fraction(N3, D3)\right) = {?}`

NUMERATOR / DENOMINATOR

First, we can simplify the problem:

`fractionReduce(N1, D1) \div fractionReduce(N2, D2) \left(fractionReduce(N1, D1) \div fractionReduce(N2, D2)\right) \div fractionReduce(N3, D3) fractionReduce(N1, D1) \div \left(fractionReduce(N2, D2) \div fractionReduce(N3, D3)\right) = {?}`

Dividing by a fraction is the same as multiply by the reciprocal of that fraction.

`\qquad = fractionReduce(N1, D1) \times fractionReduce(D2, N2) \left(fractionReduce(N1, D1) \times fractionReduce(D2, N2)\right) \div fractionReduce(N3, D3) fractionReduce(N1, D1) \div \left(fractionReduce(N2, D2) \times fractionReduce(D3, N3)\right)`

`\qquad = \dfrac{N1 / F1 \times D2 / F2-D2 / F2} {D1 / F1 \times abs(N2) / F2} \left(\dfrac{N1 / F1 \times D2 / F2-D2 / F2} {D1 / F1 \times abs(N2) / F2}\right) \div fractionReduce(N3, D3) fractionReduce(N1, D1) \div \left(\dfrac{N2 / F2 \times D3 / F3-D3 / F3} {D2 / F2 \times abs(N3) / F3}\right)`

`\qquad = fraction(SUB_N, SUB_D) \div fractionReduce(N3, D3) fractionReduce(N1, D1) \div fraction(SUB_N, SUB_D)`

`\qquad = fraction(SUB_N, SUB_D) \times fractionReduce(D3, N3) fractionReduce(N1, D1) \times fraction(SUB_D, SUB_N)`

`\qquad = \dfrac{SUB_N \times D3 / F3}{SUB_D \times N3 / F3} \dfrac{N1 / F1 \times SUB_D}{D1 / F1 \times SUB_N}`

`\qquad = fraction(NUMERATOR, DENOMINATOR)`

Simplify:

`\qquad = fractionReduce(NUMERATOR, DENOMINATOR)`

randRange(2, 3) randRange(0, 1) randFromArray([1, -1]) randFromArray([1, -1]) randFromArray([1, -1]) S1 * S2 * S3 randRange(2, 9) randRange(2, 9) randRange(1, 99) N4 * N2 * N3 randRange(0, 1) randRange(0, 1) randRange(0, 1) P2 + P3 + P4 roundTo(P2, S2 * N2 / pow(10, P2)) roundTo(P3, S3 * N3 / pow(10, P3)) roundTo(P4, S4 * N4 / pow(10, P4)) roundTo(P1, D4 * D2 * D3) NUMS === 2 || ORDER ? N1 : N1 * N3 NUMS === 2 || ORDER ? P1 : decimalPlaces(roundTo(4, D1 * D3)) NUMS === 2 || ORDER ? S1 : S1 * S3 roundTo(DIVIDEND_P, DIVIDEND_S * DIVIDEND_N / pow(10, DIVIDEND_P)) NUMS === 2 || ORDER === 0 ? N2 : N2 * N3 NUMS === 2 || ORDER === 0 ? P2 : decimalPlaces(roundTo(4, D2 * D3)) NUMS === 2 || ORDER === 0 ? S2 : S2 * S3 roundTo(DIVISOR_P, DIVISOR_S * DIVISOR_N / pow(10, DIVISOR_P)) roundTo(4, DIVIDEND_D / DIVISOR_D) 0

`D1 \div D2 (D1 \div D2) \div D3 D1 \div (D2 \div D3)`

var g1 = new Divider(DIVISOR_N, DIVIDEND_N, DIVISOR_P, DIVIDEND_P); HINTS1 = Array(g1.getNumHints());
SOLUTION

`\qquad = D1 \times (D3 \div D2)`

`\qquad = (D1 \times D3) \div D2`

`\qquad = roundTo(4, D1 * D3) \div D2`

`\qquad = D1 \div (D2 \times D3)`

`\qquad = D1 \div roundTo(4, D2 * D3)`

graph.divider = new Divider(DIVISOR_N, DIVIDEND_N, DIVISOR_P, DIVIDEND_P); graph.divider.show();
graph.divider.showHint();

Since both `DIVIDEND_D` and `DIVISOR_D` are negative, the result is positive.

A positive number divided by a negative number is a negative number.

A negative number divided by a positive number is a negative number.

`DIVIDEND_D \div DIVISOR_D = SOLUTION`

randRange(2, 3) randRange(0, 1) randFromArray([1, -1]) randFromArray([1, -1]) randFromArray([1, -1]) S1 * S2 * S3 randRange(2, 9) randRange(2, 9) randRange(1, 99) N4 * N2 * N3 randRange(1, 2) randRange(1, 2) randRange(1, 2) P2 + P3 + P4 roundTo(P2, S2 * N2 / pow(10, P2)) roundTo(P3, S3 * N3 / pow(10, P3)) roundTo(P4, S4 * N4 / pow(10, P4)) roundTo(P1, D4 * D2 * D3) NUMS === 2 || ORDER ? N1 : N1 * N3 NUMS === 2 || ORDER ? P1 : decimalPlaces(roundTo(4, D1 * D3)) NUMS === 2 || ORDER ? S1 : S1 * S3 roundTo(DIVIDEND_P, DIVIDEND_S * DIVIDEND_N / pow(10, DIVIDEND_P)) NUMS === 2 || ORDER === 0 ? N2 : N2 * N3 NUMS === 2 || ORDER === 0 ? P2 : decimalPlaces(roundTo(4, D2 * D3)) NUMS === 2 || ORDER === 0 ? S2 : S2 * S3 roundTo(DIVISOR_P, DIVISOR_S * DIVISOR_N / pow(10, DIVISOR_P)) roundTo(4, DIVIDEND_D / DIVISOR_D) 0

`roundTo(2, D1*100)\% \div roundTo(2, D2 * 100)\% (roundTo(2, D1 * 100)\% \div roundTo(2, D2 * 100)\%) \div roundTo(2, D3 * 100)\% roundTo(2, D1 * 100)\% \div (roundTo(2, D2 * 100)\% \div roundTo(2, D3 * 100)\%)`

var g1 = new Divider(DIVISOR_N, DIVIDEND_N, DIVISOR_P, DIVIDEND_P); HINTS1 = Array(g1.getNumHints());
roundTo(2, 100 * SOLUTION)`\ \%`

Convert each percentage into a decimal dividing by `100`.

`D1 \div D2 (D1 \div D2) \div D3 D1 \div (D2 \div D3)`

`\qquad = D1 \times (D3 \div D2)`

`\qquad = (D1 \times D3) \div D2`

`\qquad = roundTo(4, D1 * D3) \div D2`

`\qquad = D1 \div (D2 \times D3)`

`\qquad = D1 \div roundTo(4, D2 * D3)`

graph.divider = new Divider(DIVISOR_N, DIVIDEND_N, DIVISOR_P, DIVIDEND_P); graph.divider.show();
graph.divider.showHint();

Since both `DIVIDEND_D` and `DIVISOR_D` are negative, the result is positive.

A positive number divided by a negative number is a negative number.

A negative number divided by a positive number is a negative number.

`DIVIDEND_D \div DIVISOR_D = SOLUTION`

Convert the decimal into a percentage by multiplying by `100`.

`SOLUTION = roundTo(10, SOLUTION * 100)\%`

2 ['x', 'y', 'z'] [ randRangeWeighted(-3, 3, 0, 0.5), randRangeWeighted(-3, 3, 0, 0.5), randRangeWeighted(-3, 3, 0, 0.5)] [randRange(1, 2), randRange(1, 2), randRange(1, 2)] [ (DIGITS[0] === 1) ? randRange(1, 8) : randRange(10, 98), (DIGITS[1] === 1) ? randRange(1, 8) : randRange(10, 98), (DIGITS[2] === 1) ? randRange(1, 8) : randRange(10, 98)] [pow(10, DIGITS[0]) - 1, pow(10, DIGITS[1]) - 1, pow(10, DIGITS[2]) - 1] [ INTEGERS[0] * DENOMINATORS[0] + REPEATS[0] * (INTEGERS[0] < 0 ? -1 : 1), INTEGERS[1] * DENOMINATORS[1] + REPEATS[1] * (INTEGERS[1] < 0 ? -1 : 1), INTEGERS[2] * DENOMINATORS[2] + REPEATS[2] * (INTEGERS[2] < 0 ? -1 : 1)] [ INTEGERS[0] + (INTEGERS[0] < 0 ? -1 : 1) * REPEATS[0] / DENOMINATORS[0], INTEGERS[1] + (INTEGERS[1] < 0 ? -1 : 1) * REPEATS[1] / DENOMINATORS[1], INTEGERS[2] + (INTEGERS[2] < 0 ? -1 : 1) * REPEATS[2] / DENOMINATORS[2]] max(1, DENOMINATORS[0] / DENOMINATORS[1]) max(1, DENOMINATORS[1] / DENOMINATORS[0]) NUMERATORS[0] * F2 NUMERATORS[1] * F1 getGCD(NUMERATOR, DENOMINATOR)

`INTEGERS[0].\overline{REPEATS[0]} \div INTEGERS[1].\overline{REPEATS[1]} = {?}`

NUMERATOR / DENOMINATOR

First convert the repeating decimals to fractions.

`\begin{align*} pow(10, DIGITS[i])LETTERS[i] &= floorTo(4, DECIMALS[i] * pow(10, DIGITS[i]))...\\ LETTERS[i] &= floorTo(4, DECIMALS[i])...\end{align*}`

`\begin{align*} DENOMINATORS[i]LETTERS[i] &= NUMERATORS[i] \\ LETTERS[i] &= fraction(NUMERATORS[i], DENOMINATORS[i])\end{align*}`

So, the problem becomes:

`fraction(NUMERATORS[0], DENOMINATORS[0]) \div fraction(NUMERATORS[1], DENOMINATORS[1]) = {?}`

Dividing by a fraction is the same as multiply by the reciprocal of that fraction.

`fraction(NUMERATORS[0], DENOMINATORS[0]) \times fraction(DENOMINATORS[1], NUMERATORS[1]) = {?}`

`\phantom{fraction(NUMERATORS[0], DENOMINATORS[0]) \times fraction(NUMERATORS[1], DENOMINATORS[1])} = \dfrac{NUMERATORS[0] \times DENOMINATORS[1]}{DENOMINATORS[0] \times NUMERATORS[1]}`

`\phantom{fraction(NUMERATORS[0], DENOMINATORS[0]) \times fraction(NUMERATORS[1], DENOMINATORS[1])} = \dfrac{NUMERATORS[0] \times \cancel{DENOMINATORS[1]}F2} {\cancel{DENOMINATORS[0]}F1 \times NUMERATORS[1]}`

`\phantom{fraction(NUMERATORS[0], DENOMINATORS[0]) \times fraction(NUMERATORS[1], DENOMINATORS[1])} = fraction(NUMERATOR, DENOMINATOR)`

Simplify:

`\large{= fractionReduce(NUMERATOR / GCD, DENOMINATOR / GCD)}`