RIDDLES — FRACTIONS AND PERCENTAGES
Types of Fractions
RIDDLE 1
I am a numerical value that represents a part of a whole, a group, or a ratio.
I am written as one number over another, separated by a horizontal or diagonal line.
The number below my line shows how many equal parts something is divided into, while the number above shows how many of those parts are being considered.
I can represent a value less than one, exactly one, or more than one, depending on how my two numbers compare.
Half of a pizza, three out of four correct answers, and two servings from a recipe can all be represented using me.
Who am I?
ANSWER: Fraction
SOLUTION
A fraction expresses a part-to-whole relationship as one integer, the numerator, written over another, the denominator, separated by a fraction line. Its value is less than, equal to, or greater than one depending on whether the numerator is smaller than, equal to, or larger than the denominator.
RIDDLE 2
I am the upper part of a fraction.
I represent the number of parts being considered out of the total.
I sit above the fraction's horizontal line, or to the left of the diagonal line in a fraction like 3/4.
In the fraction 3/4, I am the number 3.
If I am zero, the entire fraction is equal to zero, no matter what the bottom number is.
Who am I?
ANSWER: Numerator
SOLUTION
The numerator is the top number of a fraction, showing how many of the equal parts named by the denominator are being taken. Since a fraction's value equals numerator divided by denominator, a numerator of zero makes the whole fraction equal to zero regardless of the denominator.
RIDDLE 3
I am the lower part of a fraction.
I represent the total number of equal parts into which a whole is divided.
I sit below the fraction's horizontal line, or to the right of the diagonal line in a fraction like 3/4.
In the fraction 3/4, I am the number 4.
I can never be equal to zero, since dividing anything into zero equal parts is undefined.
Who am I?
ANSWER: Denominator
SOLUTION
The denominator is the bottom number of a fraction, showing into how many equal parts the whole has been split. It can never be zero, because division by zero is undefined, so every valid fraction must have a nonzero denominator.
RIDDLE 4
I am a fraction where the numerator is less than the denominator. 2/5 is an example of me.
My value always lies strictly between 0 and 1 on the number line.
I represent a quantity that is less than a single whole unit.
When converted to a decimal, I always produce a value less than 1, such as 2/5 becoming 0.4.
Who am I?
ANSWER: Proper Fraction
SOLUTION
A proper fraction has a numerator smaller than its denominator, so its value always lies strictly between 0 and 1. This reflects a quantity that is less than one complete whole unit.
RIDDLE 5
I am a fraction where the numerator is greater than the denominator. 7/5 is an example of me.
My value is always greater than 1, representing more than a single whole unit.
I can always be rewritten as a mixed number, so 7/5 becomes 1 and 2/5.
When converted to a decimal, I always produce a value greater than 1, such as 7/5 becoming 1.4.
Who am I?
ANSWER: Improper Fraction
SOLUTION
An improper fraction has a numerator greater than or equal to its denominator, giving it a value of at least 1. Every improper fraction can be converted into a mixed number by dividing the numerator by the denominator.
RIDDLE 6
I am a combination of a whole number part and a proper fraction, written side by side as a single value. 3 and 1/4 is an example of me, combining the whole number 3 with the proper fraction 1/4.
I can always be converted into an improper fraction, so 3 and 1/4 becomes 13/4.
To convert me back from an improper fraction, the denominator is divided into the numerator, with the remainder becoming the new numerator placed over the original denominator.
Recipes and everyday measurements are often expressed using me rather than as an improper fraction, since I am easier to picture.
Who am I?
ANSWER: Mixed Fraction (Mixed Number)
SOLUTION
A mixed number combines a whole-number part with a proper fraction, such as 3 and 1/4. To convert it into an improper fraction, the whole number is multiplied by the denominator and added to the numerator, with the result placed over the original denominator.
RIDDLE 7
I describe two or more fractions that share the same denominator. 1/8, 3/8 and 5/8 are examples of me, since all three share a denominator of 8.
Because our bottom numbers already match, we can be added or subtracted simply by combining our numerators.
I make comparing fraction sizes straightforward, since whichever fraction has the larger numerator is automatically the larger value.
Fractions that do not share a denominator must first be converted into equivalent fractions before they can become me.
Who am I?
ANSWER: Like Fractions
SOLUTION
Like fractions share a common denominator, so their equal-sized parts allow the numerators to be added or subtracted directly. Because the parts are the same size, comparing which fraction is larger simply means comparing numerators.
RIDDLE 8
I describe two or more fractions with different denominators. 1/3 and 3/4 are examples of me, since 3 and 4 are different numbers.
We cannot be added or subtracted directly, since our parts are divided into unequal sized pieces.
Before performing arithmetic on us, each fraction must be converted into an equivalent fraction sharing a common denominator, often the least common multiple of the original denominators.
Comparing our sizes at a glance is also difficult, unlike fractions that already share the same denominator.
Who am I?
ANSWER: Unlike Fractions
SOLUTION
Unlike fractions have different denominators, so their parts are different sizes and cannot be combined directly. They must first be rewritten as equivalent fractions sharing a common denominator, typically the least common multiple of the original denominators, before adding, subtracting, or easily comparing them.
RIDDLE 9
I describe fractions that represent the same value or proportion, even though they have different numerators and denominators. 1/2, 2/4 and 3/6 are examples of me.
I am formed by multiplying or dividing both the numerator and denominator of a fraction by the same non-zero number.
Each of my fractions, when simplified fully, reduces down to the exact same fraction in its lowest terms.
I am essential for adding or subtracting fractions with unlike denominators, since one or both fractions must be rewritten as me before combining.
Who am I?
ANSWER: Equivalent Fractions
SOLUTION
Equivalent fractions represent the same value despite having different numerators and denominators, because multiplying or dividing both parts by the same nonzero number preserves the overall proportion. Reducing any of them to lowest terms always produces the identical simplest fraction.
RIDDLE 10
I am a fraction whose numerator is equal to 1.
One-half and one-third are examples of me.
My value is found simply by dividing 1 by my denominator.
As my denominator increases, my overall value becomes smaller, since the whole is being divided into more pieces.
Ancient Egyptian mathematicians expressed almost all fractions as sums of me, using distinct denominators for each unit fraction added together.
Who am I?
ANSWER: Unit Fraction
SOLUTION
A unit fraction has a numerator of exactly 1, so its value equals 1 divided by its denominator. As the denominator grows larger, the fraction represents a smaller share of the whole, since the same unit is split into more pieces.
RIDDLE 11
I am a simple, commonly used fraction that serves as a reference point for estimating or comparing other fractions.
Zero, one-quarter, one-half, three-quarters and one whole are examples of me.
I help students judge roughly where an unfamiliar fraction, like 5/9, falls on the number line by comparing it to my familiar values.
For example, since 5/9 is close to 1/2, it can quickly be estimated as roughly one-half without doing exact arithmetic.
I am especially useful for checking whether a calculated answer to a fraction problem is reasonable.
Who am I?
ANSWER: Benchmark Fraction
SOLUTION
Benchmark fractions are familiar reference values such as 0, 1/4, 1/2, 3/4 and 1 that make it easy to judge where an unfamiliar fraction sits on the number line. Comparing an unknown fraction to the nearest benchmark gives a quick, reasonable estimate without needing exact computation.
Operations on Fractions
RIDDLE 12
To add or subtract two fractions, I must first be made the same between them, usually by finding the least common multiple of the two denominators involved.
Once I am shared between two fractions, their numerators can simply be added or subtracted directly.
For example, adding 1/4 and 1/6 requires rewriting both fractions with me equal to 12, giving 3/12 + 2/12.
The smallest possible version of me, found using the least common multiple of the original denominators, keeps the resulting numbers as small and manageable as possible.
Any common multiple of the original denominators can technically serve as me, though larger choices make simplifying the final answer harder.
Who am I?
ANSWER: Common Denominator
SOLUTION
A common denominator is a shared multiple of two or more denominators that lets fractions be added or subtracted by combining numerators directly. Using the least common multiple as the common denominator keeps the numbers as small as possible, making the resulting fraction easier to simplify.
RIDDLE 13
I am the rule that says to multiply two fractions together, you multiply their numerators together and their denominators together, with no need for a common denominator.
For example, multiplying 2/3 by 3/4 gives (2 × 3)/(3 × 4), which equals 6/12, or 1/2 once simplified.
Unlike adding or subtracting fractions, applying me does not require the denominators to match beforehand.
Before multiplying, fractions can sometimes be simplified diagonally by cancelling common factors between a numerator and the opposite denominator, making the final answer easier to reduce.
Multiplying two proper fractions using me always produces a result smaller than either original fraction.
Who am I?
ANSWER: Rule for Multiplying Fractions
SOLUTION
To multiply fractions, the numerators are multiplied together and the denominators are multiplied together, with no requirement for a common denominator. Multiplying two proper fractions always produces a result smaller than either starting fraction, since each fraction scales the other down.
RIDDLE 14
I am a memorable technique for dividing fractions: keep the first fraction the same, change the division sign to multiplication, and flip the second fraction.
For example, dividing 2/3 by 1/4 becomes 2/3 × 4/1, which equals 8/3.
Flipping the second fraction means swapping its numerator and denominator to form its reciprocal.
I work because dividing by a number gives the same result as multiplying by that number's reciprocal.
I am especially helpful for students who find the formal division rule for fractions difficult to remember.
Who am I?
ANSWER: Keep, Change, Flip (Reciprocal Method for Dividing Fractions)
SOLUTION
To divide by a fraction, the first fraction is kept unchanged, the operation is changed to multiplication, and the second fraction is flipped to its reciprocal. This works because dividing by a number is mathematically equivalent to multiplying by that number's reciprocal.
RIDDLE 15
I am the fraction formed when the numerator and denominator of another fraction are swapped.
Multiplying a fraction by me gives an answer of 1.
The value of 3/5 is 5/3, since swapping the numerator and denominator produces that pair.
Whole numbers also have me, since 7 can be rewritten as 7/1, making my value 1/7.
I am not defined for zero, since swapping its numerator and denominator would require dividing by zero.
Who am I?
ANSWER: Reciprocal
SOLUTION
The reciprocal of a fraction is formed by swapping its numerator and denominator, and multiplying a number by its reciprocal always gives 1. Zero has no reciprocal, since expressing it as 0/1 and flipping it would require dividing by zero.
RIDDLE 16
When I am added to a fraction, the result is zero.
The additive inverse of 1/2 is -1/2.
I am formed simply by changing the sign of the original fraction, from positive to negative or from negative to positive.
On a number line, I am always the same distance from zero as the original fraction, but on the opposite side.
Unlike the reciprocal, which is used for division and multiplication, I am specifically the value used to cancel a number out through addition.
Who am I?
ANSWER: Additive Inverse (of a Fraction)
SOLUTION
The additive inverse of a fraction is its negative, since adding a number to its opposite always sums to zero. On a number line it sits the same distance from zero as the original fraction, but on the opposite side.
RIDDLE 17
When I am multiplied by a fraction, the result is always 1.
I am formed by swapping the numerator and denominator of the original fraction.
For the fraction 4/9, I am the fraction 9/4.
Every non-zero number has exactly one of me, but zero has none, since no number multiplied by zero can ever equal 1.
Whole numbers have me as well, since 6 can be rewritten as 6/1, giving me the value 1/6.
Who am I?
ANSWER: Multiplicative Inverse (of a Fraction)
SOLUTION
The multiplicative inverse of a fraction is its reciprocal, since multiplying a number by its reciprocal always produces 1. Every nonzero number has exactly one multiplicative inverse, but zero has none, because no number multiplied by zero can equal 1.
Connections Between Fractions, Decimals & Percentages
RIDDLE 18
I am a number consisting of a whole number part and a fractional part, separated by a special dot.
In the number 4.75, my whole number part is 4 and my fractional part is 75 hundredths.
Each digit after my dot represents tenths, hundredths, thousandths, and so on, depending on its position.
I can be terminating, ending after a fixed number of digits, or recurring, repeating a digit or group of digits forever.
I am closely linked to fractions, since any fraction can be converted into me by dividing the numerator by the denominator.
Who am I?
ANSWER: Decimal
SOLUTION
A decimal uses place value to represent a number, with a decimal point separating the whole-number part from the fractional part expressed in tenths, hundredths and beyond. Decimals may terminate after finitely many digits or recur forever, and every fraction converts to a decimal by dividing its numerator by its denominator.
RIDDLE 19
I am the dot that divides the whole number part of a decimal from its fractional part.
In the number 6.25, I sit directly between the 6 and the 2.
Digits to my left represent whole units, tens, hundreds, and so on, while digits to my right represent tenths, hundredths, and smaller parts.
Moving me one place to the right multiplies a number by 10, while moving me one place to the left divides it by 10.
In some countries, a comma is used in my place instead of a dot.
Who am I?
ANSWER: Decimal Point
SOLUTION
The decimal point separates the whole-number digits from the fractional digits in a decimal number, with place value changing by a factor of ten across each position. Shifting it one place to the right multiplies the number by 10, while shifting it one place to the left divides it by 10.
RIDDLE 20
I am a number that can be expressed as a fraction out of a hundred.
I represent the number of parts in every 100.
I am written using the symbol %, so 45 out of 100 is written as 45%.
To convert me into a decimal, my value is divided by 100, so 45% becomes 0.45.
I can also be shown visually using a 100-square grid, where each small shaded square represents one part of me.
Who am I?
ANSWER: Percentage
SOLUTION
A percentage expresses a quantity as a fraction out of 100, with the % symbol denoting "per hundred." Converting a percentage to a decimal means dividing its value by 100, so 45% becomes 0.45.
Percentage Applications
RIDDLE 21
I measure how much a value has grown, expressed as a percentage of its original value.
I am calculated as the increase divided by the original price, multiplied by 100%.
For example, if a price rises from $50 to $60, the increase of $10 divided by 50 gives me a value of 20%.
I always compare the change against the original starting value, never against the new, larger value.
I can be applied to prices, populations, measurements, or any quantity that changes in size over time.
Who am I?
ANSWER: Percentage Increase
SOLUTION
Percentage increase measures growth relative to the original value, calculated as the increase divided by the original value, multiplied by 100%. It always uses the original, smaller value as the base of comparison, never the new, larger one.
RIDDLE 22
I measure the reduction in a value, expressed as a percentage of its original value.
I am calculated as the decrease divided by the original price, multiplied by 100%.
For example, if a price falls from $80 to $60, the decrease of $20 divided by 80 gives me a value of 25%.
Like percentage increase, I always compare the change against the original starting value, not the new, smaller value.
I am commonly used to describe discounts, depreciation, or any reduction in a quantity over time.
Who am I?
ANSWER: Percentage Decrease
SOLUTION
Percentage decrease measures a reduction relative to the original value, calculated as the decrease divided by the original value, multiplied by 100%. Like percentage increase, the original value is always the base for comparison, regardless of how much smaller the new value becomes.
RIDDLE 23
I am a fee or percentage of a sale amount paid to a salesperson or agent for facilitating a sale.
I am usually calculated as a percentage rate multiplied by the total value of the sale made.
For example, a 5% rate on a $2,000 sale earns the salesperson $100.
I give salespeople a direct financial incentive to sell more or higher-value goods, since my amount rises with the sale amount.
I differ from a fixed salary, since my value changes from period to period depending on sales performance.
Who am I?
ANSWER: Commission
SOLUTION
Commission is a percentage-based payment earned by a salesperson for facilitating a sale, calculated as a rate multiplied by the value of the goods sold. Because it scales directly with sales performance, it rewards higher sales volume, unlike a fixed salary.
RIDDLE 24
I am a reduction in the original price of a product or service, often offered to encourage purchases.
I am usually expressed as either a fixed amount of money or a percentage of the original marked price.
For example, a 20% version of me on a $50 item reduces the price by $10, to $40.
Stores often offer me during sales events to clear old stock or attract more customers.
Once I am subtracted from the marked price, the amount remaining is what the customer actually pays.
Who am I?
ANSWER: Discount
SOLUTION
A discount reduces an item's original marked price, expressed either as a fixed amount or as a percentage of that price. Subtracting the discount from the marked price gives the amount the customer actually pays.
RIDDLE 25
I am the financial gain made when the selling price of a product is greater than its cost price.
I am calculated by subtracting the cost price from the selling price.
For example, an item bought for $30 and sold for $45 generates me in the amount of $15.
The larger the gap between the selling price and the cost price, the greater my value becomes.
Businesses aim to maximise me while keeping prices fair enough to attract customers.
Who am I?
ANSWER: Profit
SOLUTION
Profit is the financial gain made when an item's selling price exceeds its cost price, found by subtracting the cost price from the selling price. A wider gap between the two prices produces a larger profit.
RIDDLE 26
I am profit expressed as a percentage of the cost price.
I am calculated as profit divided by the cost price, multiplied by 100%.
For example, an item bought for $30 and sold for $45 makes a profit of $15, giving me a value of 50%.
I allow profits on items of very different prices to be compared fairly on the same scale.
Unlike plain profit, which is measured in currency, I am always measured as a percentage.
Who am I?
ANSWER: Percentage Profit
SOLUTION
Percentage profit expresses profit as a proportion of the cost price, calculated as profit divided by cost price, multiplied by 100%. Expressing profit this way allows fair comparison across items of very different prices.
RIDDLE 27
I occur when the selling price of a product is less than its cost price.
I am calculated by subtracting the selling price from the cost price.
For example, an item bought for $40 and sold for $30 generates me in the amount of $10.
I am the opposite outcome to profit, representing money given up rather than gained on a sale.
Businesses try to minimise me, sometimes accepting me deliberately to clear unwanted stock.
Who am I?
ANSWER: Loss
SOLUTION
A loss occurs when an item's selling price is lower than its cost price, found by subtracting the selling price from the cost price. It represents the opposite outcome to profit, meaning money given up rather than gained.
RIDDLE 28
I am the price at which an item was originally bought, before any profit or loss is calculated.
I represent the amount a seller had to spend to acquire or produce the item in the first place.
Both profit and loss are measured by comparing me against the eventual selling price.
If the selling price exceeds me, the seller makes a profit, but if it falls below me, the seller makes a loss.
For a shopkeeper, I might include the wholesale price paid to a supplier, plus any additional costs of preparing the item for sale.
Who am I?
ANSWER: Cost Price
SOLUTION
Cost price is the amount originally paid to acquire or produce an item, before any profit or loss is calculated. Comparing the eventual selling price against the cost price determines whether the transaction results in a profit or a loss.
RIDDLE 29
I am the price at which an item is sold to a customer.
I am compared against the cost price to determine whether a profit or a loss has been made.
If I am greater than the cost price, the difference between us is called profit.
If I am less than the cost price, the difference is instead called a loss.
I may also be lower than the marked price, when a discount has been applied before the sale is completed.
Who am I?
ANSWER: Selling Price
SOLUTION
Selling price is the amount a customer actually pays for an item, compared against the cost price to determine profit or loss. It can fall below the marked price whenever a discount has been applied before the sale.
RIDDLE 30
I am the original, labelled price of an item before any discount is applied.
I am usually the price displayed on a price tag or advertised to customers in a shop.
When a discount is subtracted from me, the result becomes the item's actual selling price.
For example, if I am $100 and a 15% discount is applied, the item sells for $85.
I am not always the same as the cost price, since a shop may set me higher to allow room for future discounts while still making a profit.
Who am I?
ANSWER: Marked Price
SOLUTION
Marked price is the original, displayed price of an item before any discount is subtracted. A shop can set the marked price above the cost price, leaving room to offer discounts while still making a profit.
Simple & Compound Interest
RIDDLE 31
I am the sum of money lent, borrowed or invested initially.
I am denoted by the letter P.
I do not include any interest that accumulates afterwards, representing only the original starting amount.
In both simple and compound interest formulas, I serve as the base amount upon which all interest is calculated.
For example, depositing $500 into a savings account means I am equal to $500 at the very start.
Who am I?
ANSWER: Principal
SOLUTION
The principal, denoted P, is the original sum of money lent, borrowed or invested, before any interest is added. It serves as the base amount from which both simple and compound interest are calculated.
RIDDLE 32
I am the extra money paid for taking money as a loan, often expressed as a percentage.
I represent the cost of borrowing money, or equivalently, the reward earned for lending or investing it.
I depend on three main factors: the principal amount, the rate at which I accumulate, and the length of time involved.
I can be calculated as simple, growing only from the original principal, or compound, growing from the principal plus previously earned amounts.
Banks pay me to savers to encourage deposits, while also charging me to borrowers for loans.
Who am I?
ANSWER: Interest
SOLUTION
Interest is the cost of borrowing money, or equivalently the reward for lending or investing it, depending on the principal, rate and time involved. It can be simple, growing only from the original principal, or compound, growing from the principal plus previously earned interest.
RIDDLE 33
I am the percentage at which a principal amount accrues interest over a period of time.
I am denoted by the letter R.
I am usually expressed on a yearly basis, so this figure is typically used directly within most interest formulas without needing conversion.
A higher value of me means interest builds up more quickly on the same principal amount.
When used in the simple interest formula PRT/100, I must be entered as a plain number, such as 5 for 5%, rather than as a decimal.
Who am I?
ANSWER: Rate (of Interest)
SOLUTION
The rate, denoted R, is the percentage at which a principal accrues interest over time, usually stated per year. A higher rate means interest builds up more quickly on the same principal amount.
RIDDLE 34
I am the duration for which a principal amount is lent, borrowed or invested, usually measured in years.
I am denoted by the letter T in interest formulas.
If a period is given in months rather than years, I must be converted by dividing the number of months by 12 before being used in a formula.
The longer my value, the more interest accumulates on a given principal, assuming the rate stays constant.
In compound interest calculations, I also determine how many times interest gets added back onto the growing principal.
Who am I?
ANSWER: Time
SOLUTION
Time, denoted T, is the duration a principal is invested or borrowed for, usually measured in years, with a period given in months converted by dividing by 12. A longer time period allows more interest to accumulate on the same principal at a given rate.
RIDDLE 35
I am a type of interest calculated only on the original principal amount, without accounting for any interest accumulated previously.
I am calculated using the formula PRT/100, where P is the principal, R is the rate, and T is the time.
Because I ignore previously earned interest, I grow by the same fixed amount every single period.
For example, $1000 invested at a rate of 5% per year earns exactly $50 in interest every year, for as long as the investment lasts.
When plotted on a graph against time, my accumulated total forms a straight line, reflecting my constant, linear growth.
Who am I?
ANSWER: Simple Interest
SOLUTION
Simple interest is calculated only on the original principal using the formula PRT/100, so it adds the same fixed amount every period. Because it never earns interest on previously earned interest, its accumulated total grows in a straight line when plotted against time.
RIDDLE 36
I am a type of interest calculated on the principal amount together with the interest already accumulated.
Unlike my simpler counterpart, I allow for exponential growth of an investment over time.
My total amount after T years can be found using the formula A = P(1 + R/100)T, where P is the principal and R is the rate.
Because each period's interest is calculated on a growing balance, the amount I add increases period after period, rather than staying fixed.
When plotted on a graph against time, my accumulated total forms a curve rather than a straight line, reflecting this accelerating growth.
Who am I?
ANSWER: Compound Interest
SOLUTION
Compound interest is calculated on the principal plus all interest already earned, using A = P(1 + R/100)^T, so its growth accelerates with each period. Because each period's interest is calculated on an ever-growing balance, the accumulated total forms a curve rather than a straight line when plotted against time.