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287
lib/problems/generators/decimal-problems.ts
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287
lib/problems/generators/decimal-problems.ts
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import type { MathProblem, Difficulty } from "../types";
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import { randomInt, randomChoice, shuffle } from "@/lib/utils";
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import { roundToDecimalPlaces, roundToSignificantFigures, toStandardForm, standardFormToKatex } from "@/lib/math/decimals";
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let counter = 0;
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function nextId() {
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return `dp-${++counter}-${Date.now()}`;
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}
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export function generateDecimalCompareOrder(difficulty: Difficulty): MathProblem {
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const count = difficulty === 1 ? 3 : difficulty === 2 ? 4 : 5;
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const values: number[] = [];
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for (let i = 0; i < count; i++) {
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const intPart = randomInt(0, difficulty === 1 ? 2 : 10);
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const decPlaces = randomInt(1, difficulty === 1 ? 2 : 3);
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const decPart = randomInt(1, Math.pow(10, decPlaces) - 1);
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const val = intPart + decPart / Math.pow(10, decPlaces);
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values.push(Math.round(val * 1000) / 1000);
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}
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const sorted = [...values].sort((a, b) => a - b);
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const shuffled = shuffle(values);
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return {
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id: nextId(),
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prompt: `\\text{Order from smallest to largest: } ${shuffled.join(", ")}`,
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answer: { kind: "decimal", value: 0 },
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hints: [
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"Compare the whole number part first",
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"If the whole parts are equal, compare tenths, then hundredths",
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"Writing them with the same number of decimal places can help",
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],
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steps: sorted.map((v, i) => ({
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explanation: `Position ${i + 1}: ${v}`,
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})),
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};
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}
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export function generateDecimalRounding(difficulty: Difficulty): MathProblem {
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const value = randomInt(100, 99999) / (difficulty === 1 ? 100 : 1000);
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const roundType = randomChoice(
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difficulty === 1
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? ["whole", "1dp"] as const
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: difficulty === 2
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? ["1dp", "2dp", "whole"] as const
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: ["1sf", "2sf", "3sf"] as const,
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);
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let answer: number;
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let instruction: string;
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switch (roundType) {
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case "whole":
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answer = Math.round(value);
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instruction = "the nearest whole number";
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break;
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case "1dp":
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answer = roundToDecimalPlaces(value, 1);
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instruction = "1 decimal place";
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break;
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case "2dp":
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answer = roundToDecimalPlaces(value, 2);
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instruction = "2 decimal places";
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break;
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case "1sf":
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answer = roundToSignificantFigures(value, 1);
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instruction = "1 significant figure";
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break;
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case "2sf":
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answer = roundToSignificantFigures(value, 2);
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instruction = "2 significant figures";
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break;
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case "3sf":
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answer = roundToSignificantFigures(value, 3);
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instruction = "3 significant figures";
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break;
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}
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return {
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id: nextId(),
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prompt: `\\text{Round } ${value} \\text{ to ${instruction}}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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"Look at the digit to the right of where you're rounding",
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"If it's 5 or more, round up. If it's less than 5, round down",
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],
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steps: [
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{ explanation: `Rounding ${value} to ${instruction}` },
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{ explanation: `Answer: ${answer}` },
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],
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};
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}
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export function generateStandardForm(difficulty: Difficulty): MathProblem {
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let value: number;
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if (difficulty === 1) {
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value = randomChoice([7438, 1578, 62, 93000, 5200, 340]);
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} else if (difficulty === 2) {
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value = randomChoice([0.086, 0.748, 0.0055, 0.34, 0.007]);
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} else {
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value = randomChoice([12436.3, 0.00092, 504000, 0.0000067, 83100]);
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}
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const sf = toStandardForm(value);
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return {
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id: nextId(),
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prompt: `\\text{Express } ${value} \\text{ in standard form}`,
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answer: { kind: "standardForm", coefficient: sf.coefficient, exponent: sf.exponent },
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hints: [
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"Move the decimal point so there's one non-zero digit before it",
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"Count how many places you moved the decimal",
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value >= 1
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? "Number is ≥ 1, so the power is positive"
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: "Number is < 1, so the power is negative",
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],
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steps: [
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{ explanation: `Move the decimal point to get a number between 1 and 10` },
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{ explanation: `Coefficient: ${sf.coefficient}` },
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{ explanation: `The decimal moved ${Math.abs(sf.exponent)} places, power = ${sf.exponent}` },
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{ explanation: "Answer", math: standardFormToKatex(sf.coefficient, sf.exponent) },
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],
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};
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}
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export function generateDecimalAddSubtract(difficulty: Difficulty): MathProblem {
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const operation = randomChoice(["add", "subtract"] as const);
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const op = operation === "add" ? "+" : "-";
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const places = difficulty === 1 ? 1 : difficulty === 2 ? 2 : 3;
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const a = randomInt(100, 9999) / Math.pow(10, places);
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const b = randomInt(100, 9999) / Math.pow(10, places);
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const [first, second] = operation === "subtract" ? [Math.max(a, b), Math.min(a, b)] : [a, b];
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const answer = roundToDecimalPlaces(
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operation === "add" ? first + second : first - second,
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places,
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);
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return {
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id: nextId(),
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prompt: `${first} ${op} ${second}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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"Line up the decimal points",
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"Add zeros to fill empty places",
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operation === "add" ? "Add column by column from right to left" : "Subtract column by column, borrow if needed",
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],
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steps: [
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{ explanation: `Line up: ${first} ${op} ${second}` },
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{ explanation: `= ${answer}` },
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],
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};
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}
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export function generateDecimalMultiplyDivide(difficulty: Difficulty): MathProblem {
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const operation = randomChoice(["multiply", "divide"] as const);
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if (difficulty === 1) {
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// By powers of 10
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const power = randomChoice([10, 100, 1000]);
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const base = randomInt(1, 999) / 100;
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if (operation === "multiply") {
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const answer = roundToDecimalPlaces(base * power, 10);
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return {
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id: nextId(),
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prompt: `${base} \\times ${power}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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`When multiplying by ${power}, move decimal ${Math.log10(power)} place(s) right`,
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],
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steps: [
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{ explanation: `Move decimal point ${Math.log10(power)} place(s) to the right` },
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{ explanation: `= ${answer}` },
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],
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};
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} else {
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const answer = roundToDecimalPlaces(base / power, 10);
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return {
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id: nextId(),
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prompt: `${base} \\div ${power}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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`When dividing by ${power}, move decimal ${Math.log10(power)} place(s) left`,
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],
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steps: [
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{ explanation: `Move decimal point ${Math.log10(power)} place(s) to the left` },
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{ explanation: `= ${answer}` },
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],
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};
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}
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}
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// By whole numbers or decimals
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const a = randomInt(10, 999) / 10;
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const b = difficulty === 2 ? randomInt(2, 12) : randomInt(10, 99) / 10;
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if (operation === "multiply") {
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const answer = roundToDecimalPlaces(a * b, 4);
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return {
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id: nextId(),
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prompt: `${a} \\times ${b}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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"Multiply as if there are no decimal points",
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"Count total decimal places in both numbers",
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"Place the decimal point in your answer",
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],
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steps: [
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{ explanation: `${a} × ${b}` },
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{ explanation: `= ${answer}` },
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],
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};
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} else {
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const answer = roundToDecimalPlaces(a / b, 4);
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return {
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id: nextId(),
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prompt: `${a} \\div ${b}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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"Place decimal point in answer directly above the one in the dividend",
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"Divide as normal",
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],
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steps: [
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{ explanation: `${a} ÷ ${b}` },
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{ explanation: `= ${answer}` },
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],
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};
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}
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}
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export function generateDecimalConversion(difficulty: Difficulty): MathProblem {
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const direction = randomChoice(["toFraction", "toDecimal"] as const);
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if (direction === "toFraction") {
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const decimal = randomChoice(
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difficulty === 1
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? [0.5, 0.25, 0.75, 0.1, 0.2, 0.4]
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: [0.125, 0.375, 0.625, 0.875, 0.15, 0.35, 0.45],
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);
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const den = decimal === 0.5 ? 2 : decimal === 0.25 || decimal === 0.75 ? 4 : [0.125, 0.375, 0.625, 0.875].includes(decimal) ? 8 : 100;
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const num = Math.round(decimal * den);
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const g = gcdSimple(num, den);
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return {
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id: nextId(),
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prompt: `\\text{Convert } ${decimal} \\text{ to a fraction}`,
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answer: { kind: "fraction", numerator: num / g, denominator: den / g },
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hints: [
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`Write as ${num}/${den}`,
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"Simplify by dividing numerator and denominator by their GCD",
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],
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steps: [
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{ explanation: `${decimal} = ${num}/${den}` },
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{ explanation: `Simplified: ${num / g}/${den / g}` },
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],
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};
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} else {
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const den = randomChoice(difficulty === 1 ? [2, 4, 5, 10] : [3, 8, 20, 25, 40, 50]);
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const num = randomInt(1, den - 1);
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const answer = roundToDecimalPlaces(num / den, 6);
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return {
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id: nextId(),
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prompt: `\\text{Convert } \\frac{${num}}{${den}} \\text{ to a decimal}`,
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answer: { kind: "decimal", value: answer },
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hints: [
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`Divide ${num} by ${den}`,
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"Or find an equivalent fraction with denominator 10, 100, etc.",
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],
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steps: [
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{ explanation: `${num} ÷ ${den} = ${answer}` },
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],
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};
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}
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}
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function gcdSimple(a: number, b: number): number {
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a = Math.abs(a);
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b = Math.abs(b);
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while (b) {
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[a, b] = [b, a % b];
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}
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return a;
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}
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