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In-depth guides on formulas, worked examples, and practical methods for maths, packaging, finance, fitness, home, and content creation problems.

Bar chart comparing expected annual profit of $360,000 under a true 50% margin against actual annual profit of $240,000 when a 50% markup was applied instead, with a $120,000 gap called out between the two bars

July 31, 2026·13 min read

Markup vs Margin: Why a 50% Markup Isn't a 50% Profit Margin

A 50% markup gives you a 33% margin, not 50%. What that gap actually costs at scale, in a contract, and which one to use depending on what you're doing.

Personal Finance
Horizontal bar chart showing all 16 routes from this comparison sorted from shortest, Dallas to Houston at 225 miles, to longest, San Diego to Hawaii at 2,610 miles, color-coded by region

July 29, 2026·12 min read

Flight Distance Between Major US Cities

How far is it, in the air, between major US cities? A straight-line flight distance chart for popular domestic and cross-border routes, plus 5 worked examples.

Travel
Timeline showing the 2026 tariff legal history from the struck-down IEEPA tariffs to the July 24 Section 301 tariffs taking effect on 60 countries

July 27, 2026·14 min read

Trump Just Taxed 99% of US Imports From 60 Countries: Here's What It Means

A new 10-12.5% tariff took effect July 24, 2026 on imports from 60 countries. What changed, who's affected, and how to calculate the real cost.

Personal Finance
Stacked bar chart comparing shelf price buildup before and after a 50 percent tariff, showing import cost, domestic costs, retail margin, and sales tax, with the final price rising 25 percent despite a 50 percent tariff rate

July 22, 2026·13 min read

Do Tariffs Really Raise Prices That Much? The Math Behind the Sticker Shock

A 50% tariff does not mean a 50% price hike. The actual cost-plus math retailers use to turn a tariff into a shelf price, and how to run the numbers yourself.

Personal Finance
Comparison of aspect ratios and pixel dimensions across YouTube 16:9, Instagram Feed 4:5, LinkedIn square 1:1, and TikTok/Reels/Shorts 9:16

July 16, 2026·12 min read

What Aspect Ratio Should You Use for YouTube, Instagram, TikTok & Shorts?

The exact 2026 aspect ratio and pixel size for every major platform, the safe zones where UI actually covers your video, and how to shoot once and crop for all of them.

Content Creation
A table identifying three concrete stain types, oil and grease, paint, and rust, with what each looks like, the right approach, and what to reach for

July 16, 2026·12 min read

How to Remove Stains from Concrete: Paint, Oil & Rust

A no-nonsense guide to removing paint, oil, and rust stains from concrete: the method that works for each, honest expectations, and the safety steps that matter.

Home & DIY
A bar chart showing where a perfect 4.0 fifteen-credit semester lands a 2.8 GPA, rising to 3.40 with only 15 prior credits but just 2.93 once 120 credits are completed

July 12, 2026·12 min read

What GPA Do You Need This Semester to Raise Your Cumulative GPA?

The honest math on raising your GPA in one semester: the exact term GPA you need to hit your target, why it gets harder with more credits, and when a goal is not possible this term.

Maths
Infographic showing a rate moving from 4 percent to 5 percent, described three ways: plus 1 percentage point, plus 25 percent relative change, and plus 100 basis points

July 10, 2026·12 min read

Percentage Points vs Percent Change: The Difference Explained

Rates rose from 4% to 5%: a 1% rise or a 25% rise? Neither and both, depending on the word you use. Percentage points, percent change, and basis points explained with five worked examples.

Maths
A cheat sheet card showing three official hydration numbers: 8 ounces every 15 to 20 minutes, 32 ounces per hour as the target, and 48 ounces per hour as the ceiling

July 8, 2026·11 min read

How Much Water Should You Drink in Hot Weather?

The official CDC, NIOSH, and OSHA numbers: 8 oz every 15 to 20 minutes, about 32 oz per hour, and a 48 oz per hour ceiling. Adjustments for workers, athletes, kids, and older adults, with five worked examples.

Fitness
A number line from 8:00 to 8:15 split at the seven-and-a-half minute midpoint, showing that clock-ins from 1 to 7 minutes past round down and 8 to 14 minutes past round up

July 6, 2026·12 min read

The 7-Minute Rule Explained: Payroll Time Rounding & FLSA Compliance

Is rounding your timecard legal? How the 7-minute rule works, the three FLSA-approved rounding increments, when rounding becomes wage theft, and how to check your own paycheck.

Personal Finance
Line chart showing the US personal savings rate collapsing from a 2020 spike of 32 percent to 2.6 percent in April 2026

July 4, 2026·13 min read

The 2026 Savings Squeeze: How Much Americans Are Really Saving

The US personal savings rate has fallen to 2.6%, less than a third of its long-run average. What the data shows, who feels it most, and a realistic plan to rebuild the habit with five worked examples.

Personal Finance
Four small coordinate graphs showing the four types of slope: positive, negative, zero, and undefined

July 1, 2026·11 min read

Positive vs Negative Slope: How to Tell the Difference

A positive slope rises left to right, a negative slope falls. Four ways to read the sign, real-world examples, five worked problems, common mistakes, and an 11-question FAQ.

Algebra
Diagram showing where to measure your waist, midway between the lowest rib and top of the hip bone, with Asian waist thresholds of 90 cm for men and 80 cm for women

June 29, 2026·12 min read

Waist Size and Waist-to-Height Ratio for Asians: The Number BMI Misses

Asian waist risk thresholds are lower than standard charts show: 90 cm for men, 80 cm for women. Keep your ratio below 0.5. Five worked examples, correct measurement technique, and 11 FAQs.

Fitness
Diagram showing that cubing a side length gives the volume and taking the cube root of the volume gives the side length back

June 26, 2026·10 min read

How to Find the Side Length of a Cube from Its Volume

Use s = ∛V to reverse the cube volume formula. Three methods: calculator shortcut, bracket estimation, and prime factoring. Includes a reference table, five worked examples, and 11 FAQs.

Geometry
Three cones of the same base radius and height filling one cylinder exactly, showing a cone is one-third of its matching cylinder

June 19, 2026·10 min read

Why a Cone Is Exactly One-Third of a Cylinder

The ⅓πr²h formula is not a coincidence: cross-sectional area shrinks with the square of the radius, not linearly. Includes the calculus proof, Cavalieri’s principle, 2,400 years of history, and five worked examples.

Geometry
Vertical cylindrical tank labelled with diameter 2 m, height 3 m, and the formula V equals pi r squared h

June 19, 2026·11 min read

How Much Water Does a Cylindrical Tank Hold? Litres and Gallons Explained

Use V = πr²h to find litres, US gallons, or imperial gallons for any cylinder. Includes a dipstick reference table, five worked examples, and the partly-filled horizontal tank formula.

Volume
Three layers of spheres inside a cylinder with radius R and height H labelled

June 17, 2026·10 min read

Sphere Packing in a Cylinder: How to Calculate the Optimal Radius and Height

Use the radius ratio table to find spheres per layer, then divide height by diameter to count layers. Worked examples for r = 7.5 h = 14.5, r = 10 h = 11, and more.

Sphere Packing
27 spheres in a 12 × 12 × 12 cm cube

June 15, 2026·8 min read

How to Find the Optimal Box Dimensions to Pack 27 Spheres

Guessing the box size for 27 spheres? See the full proof, the formula for any count (34, 36, 64, and beyond), and a calculator that solves it instantly.

Sphere Packing