Skip to content

Learn the math

Guides & articles

A growing library of practical reads behind the calculators.

Financial guide3 min read

How Mortgage Payments Work: A Worked Example

A mortgage estimate is useful only when you know what it includes. This guide follows a fixed-rate, fully amortizing loan with equal monthly payments. It explains the calculation used by our Mortgage Calculator; it is not a lender quote or a recommendation to borrow.

Start with the amount you actually borrow

Subtract the down payment from the home price. A $250,000 home with a $50,000 down payment gives a $200,000 starting loan balance. The down payment is an upfront cost, not part of the monthly principal-and-interest payment. Financing additional fees would increase the balance, so include them when comparing a real offer.

Enter the contractual annual interest rate for the payment calculation. A disclosed APR can include finance charges and is not always the same rate used to compute a loan payment. Our simple payment model does not reproduce every lender convention or fee schedule.

Convert the rate and term before using the formula

For an annual interest rate of 6%, the monthly decimal rate is r = 0.06 / 12 = 0.005. A 30-year term has n = 30 × 12 = 360 payments. The formula is M = P × r(1+r)^n / ((1+r)^n − 1). With P = $200,000, the estimated payment is $1,199.10 per month.

At a zero interest rate, use M = P / n instead; substituting zero directly into the interest-bearing fraction would divide by zero. This example assumes the same rate and payment frequency throughout the term.

See what happens in the first payment

The first month’s interest is $200,000 × 0.005 = $1,000. About $199.10 of the $1,199.10 payment reduces principal, leaving approximately $199,800.90. In the next month, interest is calculated on that lower balance, so a slightly larger share of the same payment reduces principal.

This is amortization. It explains why a fixed payment can contain different amounts of principal and interest over time. Small differences in rounding and the timing of the first payment can make a lender’s schedule differ from a simplified illustration.

Distinguish the loan payment from housing costs

The displayed figure covers principal and interest. Depending on your location and loan, property taxes, homeowners insurance, mortgage insurance, association fees, and other costs may be additional. Taxes and insurance collected through escrow can change even when the principal-and-interest amount stays fixed.

For a budget illustration, adding $300 in monthly taxes and $100 in insurance to $1,199.10 gives $1,599.10 before other costs. These two added amounts are assumptions for the example, not local price estimates.

Compare one variable at a time

Hold the loan balance and rate constant when comparing terms. A shorter term generally requires a larger monthly payment and can reduce interest paid over the full schedule. A lower monthly payment by itself does not prove a cheaper loan if the term, fees, or amount borrowed has also changed.

Extra principal payments can shorten repayment, but confirm how the lender applies them and whether contractual restrictions or charges apply. The Mortgage Calculator does not model extra payments; use the relevant repayment tool and confirm the actual treatment with the lender.

A quick check before relying on the estimate

Check the price and down payment units, annual rate, number of years, and whether extra costs are included. Keep a record of your assumptions alongside the result. Compare the estimate with the lender’s written figures instead of treating the calculator as an approval decision.

Sources and further reading

Updated · Ghreeb Calculator

Health guide3 min read

Calories, Resting Energy, and Macros: Reading an Estimate

Energy calculators turn a few measurements into a rough estimate. They do not measure your metabolism, diagnose a condition, or determine the right intake for you. This guide explains the arithmetic and the uncertainty, rather than prescribing a calorie target or promising a change in body weight.

Resting energy is only one part of a day

Your body uses energy at rest as well as during daily activity. BMR and resting energy expenditure are related concepts measured under different conditions; a web calculator is estimating them rather than carrying out either laboratory measurement. The Mifflin–St Jeor equation used here was developed to predict resting energy expenditure.

The equation uses weight, height, age, and a sex-specific constant. A result does not account for every difference in body composition, health, medication, or individual physiology. Treat extra decimal places as display precision, not proof of medical accuracy.

Follow the units in a worked example

For the equation’s male constant, enter weight in kilograms, height in centimeters, and age in years: 10 × weight + 6.25 × height − 5 × age + 5. Using 70 kg, 175 cm, and 30 years gives 700 + 1,093.75 − 150 + 5 = 1,648.75 kcal/day as an illustrative resting-energy estimate.

The equation’s female constant is −161 instead of +5. Its use of two constants is a limitation of this predictive model, not a complete description of everyone’s physiology. Confirm unit conversions carefully before interpreting any result.

An activity multiplier adds another assumption

Our calculator multiplies the resting estimate by an activity factor to approximate daily energy expenditure. If a factor of 1.2 is selected for the example above, the arithmetic gives 1,978.5 kcal/day. This is an illustration of the selected factor, not a recommended intake.

Different activity choices can change the estimate substantially. Work, walking, exercise, and changes over time all matter; a single category cannot capture an entire lifestyle. Do not assume that two tools will agree when their activity definitions or underlying models differ.

What a macro calculation tells you

The familiar general energy factors are about 4 kcal per gram of protein, 4 per gram of carbohydrate, and 9 per gram of fat. For an arithmetic example, 25 g protein, 30 g carbohydrate, and 10 g fat contribute about 100 + 120 + 90 = 310 kcal under those factors.

This does not mean those quantities are a suitable meal or daily allocation for you. Food labeling can use more detailed factors and rounding, and a macro total does not describe fiber, micronutrients, food quality, or personal dietary needs.

Why a fixed deficit is not a weight-loss promise

A static subtraction from an estimated daily expenditure does not establish how quickly an individual will lose weight. Body weight, activity, and energy needs change, so a universal claim that a particular deficit always produces the same weekly change is misleading. We do not use such a promise in this guide.

A qualified clinician or dietitian can interpret estimates in context. These simplified adult tools are not a basis for setting diets for children, pregnancy or breastfeeding, eating disorders, or conditions requiring individualized nutrition care.

Use the result to understand the model

Check units first, record the activity category, and compare results only when the same inputs and assumptions are used. Read the formula beside the tool. If a result is being used for a health decision, take the assumptions to an appropriate professional rather than changing treatment or food intake from the number alone.

Sources and further reading

Updated · Ghreeb Calculator

Math guide3 min read

Percentages in Everyday Life: Six Worked Examples

A percentage means “per hundred,” but the important question is: a percentage of which number? Most mistakes come from using the wrong base or combining changes that apply to different amounts. These examples show the steps so that you can check a calculator result independently.

1. Find a percentage of an amount

To find 20% of $45, convert the percentage to a decimal and multiply: 0.20 × 45 = $9. If this is a discount, subtract the discount from the original price: $45 − $9 = $36. The discount amount and the final price answer different questions.

Use the original amount as the base for this calculation. Enter 20 in a field labeled percent; enter 0.20 only when working directly with a decimal-rate formula. Mixing these two forms causes a hundredfold error.

2. Express one amount as a percentage of another

If 18 of 72 items are complete, divide the completed amount by the total and multiply by 100: 18 / 72 × 100 = 25%. Here 72 is the reference total, not the number remaining.

A zero reference total makes this percentage undefined. The right response is to check what is being measured, not to report 0% or an infinite percentage as though it were an ordinary result.

3. Measure a change from the starting value

A price moving from $80 to $92 changes by $12. Divide by the starting price: (92 − 80) / 80 × 100 = 15%. Reversing the comparison changes the base: the decrease from $92 to $80 is about 13.04%, not 15%.

Percentage change needs a meaningful starting value. With a zero starting value it is undefined. With negative starting values, a mechanically computed result can be hard to interpret; explain the amounts and context rather than relying on a percentage label alone.

4. Keep percentage points separate from percent change

An interest rate moving from 4% to 5% increases by 1 percentage point. Its relative increase is (5 − 4) / 4 × 100 = 25%. Both descriptions can be correct, but they describe different quantities.

This distinction matters when reading rates, survey results, or conversion figures. Write “percentage points” for the arithmetic difference between two percentages, and identify the starting percentage when describing a relative change.

5. Apply successive changes to the updated amount

A $100 item discounted by 20% becomes $80. Adding 10% tax to that discounted amount adds $8, producing $88. The combined factor is 0.80 × 1.10 = 0.88; subtracting 20 and adding 10 as though they share a $100 base would incorrectly give $90.

Two successive 10% discounts give $100 × 0.90 × 0.90 = $81, a total reduction of 19%. They do not give a 20% reduction. When copying this example to a real bill, check which charges the seller actually applies and to which amounts.

6. Work backward from a final amount

If a price is $120 after a 20% increase, divide by 1.20 to recover the original: $120 / 1.20 = $100. Subtracting 20% of $120 would give $96 because that subtraction uses the new price as its base.

Before entering values, write down the known amount, the percentage, and the reference base. For money, keep more precision during intermediate steps and round the final displayed amount; a bill may follow its own line-by-line rounding rules.

Updated · Ghreeb Calculator

Put it into practice

Every guide pairs with a free calculator.

Browse all calculators