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$Hank – math expert in math-master.org$

Hank

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Pre-Algebra • Algebra • Trigonometry • Statistics • Solid Geometry • Calculus

When I was in elementary and middle school, I did not feel the importance and beauty of mathematics, but when I joined the secondary stage, I began to feel the beauty and importance of mathematics, and I excelled in mathematics. I joined the College of Education, Department of Mathematics, and now I am a mathematics teacher. Then after that, I got a master's degree and a doctorate in curricula and methods of teaching mathematics.

$Math diplomas Hank – math expert in math-master.org, diplomas 1$

Author's latest answers

The diagonal of the TV is 140cm. The ratio of the height to the length of the screen is 9:16. Can the TV fit lengthwise between two shelves that are 120cm apart?

How old is Marko if he was 12 years old 3 years ago and is now 2025?

What is the area A of the rectangle with sides a = 1.3 dm; b = 0.3 m in m²?

Cost $125 markup$ 23.75 find the markup rate

Find the selling price : cost to store is 80$ markup is 60%

-1=v/2-3

A company survey found that 2.5% of its work force held an advanced college degree. If the company had 110 employees with an advanced degree how much employees did the company have?

حل النظام open curly brackets table attributes columnalign left end attributes row cell 4 x minus 5 y equals 9 end cell row cell negative 8 x plus 10 y equals 18 end cell end table close

The measures of two sides are given. Between what two numbers must the third side fall. 5 and 8

Prove that 3 $5-2x$ =7x+6 has a unique rational solution.

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^2 + l^2 = 140^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Pojednostavljujući i rješavajući za \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>l\u003C/math-field>\u003C/math-field> :\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{81}{256}l^2 + l^2 = 19600 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{337}{256}l^2 = 19600 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> l^2 = 19600 \\cdot \\frac{256}{337} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>l=\\sqrt{14867.96}\\approx122.02\\text{ cm}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Provjerom, duljina \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>l\\approx122.02\\text{ cm}\u003C/math-field>\u003C/math-field> \u003Cdiv>, premašuje razmak između polica od 120 cm, pa televizor ne može stati. \u003Cbr>\u003Cbr>\u003C/div>\u003Cdiv>Since \u003Cstrong data-start=\"923\" data-end=\"944\">\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>122.1\u003C/math-field>\u003C/math-field> cm > \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120\u003C/math-field>\u003C/math-field> cm\u003C/strong>, the TV \u003Cstrong data-start=\"953\" data-end=\"963\">cannot\u003C/strong> fit between the two shelves.\u003C/div>",829,166,"the-diagonal-of-the-tv-is-140cm-the-ratio-of-the-height-to-the-length-of-the-screen-is-9-16-can-the-tv-fit-lengthwise-between-two-shelves-that-are-120cm-apart",{"id":55,"category":47,"text_question":56,"photo_question":49,"text_answer":57,"step_text_answer":8,"step_photo_answer":8,"views":58,"likes":59,"slug":60},537974,"How old is Marko if he was 12 years old 3 years ago and is now 2025?","Solution:\u003Cbr />\n1. Determine how many years have passed since the time when Marko was 12.\u003Cbr />\n - Current year: 2025\u003Cbr />\n - Three years ago: 2025 - 3 = 2022\u003Cbr />\n\u003Cbr />\n2. Use the information that Marko was 12 years old in 2022 to find his age in 2025.\u003Cbr />\n - Marko was 12 years old in 2022.\u003Cbr />\n - Number of years from 2022 to 2025: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2025 - 2022 = 3\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate Marko's age in 2025:\u003Cbr />\n - Marko's age in 2025: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 + 3 = 15\u003C/math-field>\u003C/math-field>",958,192,"how-old-is-marko-if-he-was-12-years-old-3-years-ago-and-is-now-2025",{"id":62,"category":47,"text_question":63,"photo_question":49,"text_answer":64,"step_text_answer":8,"step_photo_answer":8,"views":65,"likes":66,"slug":67},537968,"What is the area A of the rectangle with sides\n a = 1.3 dm; b = 0.3 m in m²?","1. Um die Fläche eines Rechtecks zu berechnen, verwenden wir die Formel: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = a \\times b \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Wir konvertieren \\( a = 1.3 \\, \\text{dm} \\) in Meter:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> a = 1.3 \\, \\text{dm} = 0.13 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Die Seite \\( b \\) ist bereits in Metern gegeben:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> b = 0.3 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Wir berechnen die Fläche:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 0.13 \\, \\text{m} \\times 0.3 \\, \\text{m} = 0.039 \\, \\text{m}^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Der Flächeninhalt des Rechtecks ist \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 0.039 \\, \\text{m}^2 \u003C/math-field>\u003C/math-field>",285,57,"what-is-the-area-a-of-the-rectangle-with-sides-a-1-3-dm-b-0-3-m-in-m",{"id":69,"category":47,"text_question":70,"photo_question":49,"text_answer":71,"step_text_answer":8,"step_photo_answer":8,"views":72,"likes":73,"slug":74},537873,"Cost $125 markup $23.75 find the markup rate","To find the markup rate, use the following formula: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\text{Markup Rate} = \\left( \\frac{\\text{Markup}}{\\text{Cost}} \\right) \\times 100 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Here, the cost is $125 and the markup is $23.75.\u003Cbr>\u003Cbr>Substituting the values into the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\text{Markup Rate} = \\left( \\frac{23.75}{125} \\right) \\times 100 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Calculate the division:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{23.75}{125} = 0.19 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Multiply by 100 to get the percentage:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 0.19 \\times 100 = 19\\% \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the markup rate is 19%.",1297,259,"cost-125-markup-23-75-find-the-markup-rate",{"id":76,"category":47,"text_question":77,"photo_question":49,"text_answer":78,"step_text_answer":8,"step_photo_answer":8,"views":79,"likes":80,"slug":81},537856,"Find the selling price : cost to store is 80$ markup is 60%","Solution: \u003Cbr />\n\u003Cbr />\n1. Given:\u003Cbr />\n - Cost price (CP) = USD 80\u003Cbr />\n - Markup percentage = 60%\u003Cbr />\n \u003Cbr />\n2. Calculate the amount of markup:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Markup amount} = \\text{CP} \\times \\frac{\\text{Markup percentage}}{100}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Markup amount} = 80 \\times \\frac{60}{100}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Markup amount} = 80 \\times 0.6\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Markup amount} = 48 \\, \\text{USD}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Determine the selling price by adding the markup amount to the cost price:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Selling Price (SP)} = \\text{CP} + \\text{Markup amount}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{SP} = 80 + 48\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{SP} = 128 \\, \\text{USD}\u003C/math-field>\u003C/math-field>",1257,251,"find-the-selling-price-cost-to-store-is-80-markup-is-60",{"id":83,"category":47,"text_question":84,"photo_question":49,"text_answer":85,"step_text_answer":8,"step_photo_answer":8,"views":86,"likes":87,"slug":88},537842,"-1=v/2-3","1. Start with the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> -1 = \\frac{v}{2} - 3 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Add 3 to both sides to isolate the term with the variable: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> -1 + 3 = \\frac{v}{2} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the left side: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2 = \\frac{v}{2} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Multiply both sides by 2 to solve for \\( v \\): \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = 2 \\times 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = 4 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nAnswer: \\( v = 4 \\)",668,134,"1-v-2-3",{"id":90,"category":47,"text_question":91,"photo_question":49,"text_answer":92,"step_text_answer":8,"step_photo_answer":8,"views":93,"likes":94,"slug":95},537838,"A company survey found that 2.5% of its work force held an advanced college degree. If the company had 110 employees with an advanced degree how much employees did the company have?","1. Identify that \\(2.5\\%\\) of the total number of employees, denoted as \\(x\\), is equal to \\(110\\).\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{2.5}{100} \\times x = 110 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the percentage:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{2.5}{100} = \\frac{1}{40} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Substitute the simplified value back into the equation:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1}{40} \\times x = 110 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve for \\(x\\) by multiplying both sides by \\(40\\):\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 110 \\times 40 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Calculate to find:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 4400 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Therefore, the total number of employees is \\(4400\\).",892,178,"a-company-survey-found-that-2-5-of-its-work-force-held-an-advanced-college-degree-if-the-company-had-110-employees-with-an-advanced-degree-how-much-employees-did-the-company-have",{"id":97,"category":47,"text_question":98,"photo_question":49,"text_answer":99,"step_text_answer":8,"step_photo_answer":8,"views":100,"likes":101,"slug":102},537819,"حل النظام open curly brackets table attributes columnalign left end attributes row cell 4 x minus 5 y equals 9 end cell row cell negative 8 x plus 10 y equals 18 end cell end table close","Solution:\u003Cbr />\n1. Given system of equations:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x - 5y = 9\u003C/math-field>\u003C/math-field>\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-8x + 10y = 18\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Choose a method to solve the system. Here, we will use the elimination method:\u003Cbr />\n * Multiply the first equation by 2 to match the coefficients of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(4x - 5y = 9) \\times 2 \\Rightarrow 8x - 10y = 18\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Now add this equation to the second equation to eliminate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8x - 10y = 18\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-8x + 10y = 18\u003C/math-field>\u003C/math-field>\u003Cbr />\n ----------------------------------------\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0 = 36\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The equation \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0 = 36\u003C/math-field>\u003C/math-field> is false, indicating there is no solution.\u003Cbr />\n\u003Cbr />\nTherefore, the system of equations is inconsistent.\u003Cbr />\n\u003Cbr />",851,170,"open-curly-brackets-table-attributes-columnalign-left-end-attributes-row-cell-4-x-minus-5-y-equals-9-end-cell-row-cell-negative-8-x-plus-10-y-equals-18-end-cell-end-table-close",{"id":104,"category":47,"text_question":105,"photo_question":49,"text_answer":106,"step_text_answer":8,"step_photo_answer":8,"views":107,"likes":108,"slug":109},537795,"The measures of two sides are given. Between what two numbers must the third side fall.\n\n5 and 8","Solution:\u003Cbr />\n1. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Therefore, the following inequalities must be satisfied:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a + b > c\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a + c > b\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b + c > a\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Assign the given side lengths to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b\u003C/math-field>\u003C/math-field>, making \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 5\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b = 8\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Apply the triangle inequality theorem:\u003Cbr />\n- First inequality: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5 + 8 > c\u003C/math-field>\u003C/math-field> gives us \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c \u003C 13\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Second inequality: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c + 5 > 8\u003C/math-field>\u003C/math-field> gives us \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c > 3\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Third inequality is redundant as \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c + 8 > 5\u003C/math-field>\u003C/math-field> is always true for positive values of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n4. Therefore, the third side (\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field>must satisfy:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3 \u003C c \u003C 13\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Conclusion: The third side must be between 3 and 13.",716,143,"the-measures-of-two-sides-are-given-between-what-two-numbers-must-the-third-side-fall-5-and-8",{"id":111,"category":47,"text_question":112,"photo_question":49,"text_answer":113,"step_text_answer":8,"step_photo_answer":8,"views":114,"likes":115,"slug":116},537737,"Prove that 3(5-2x)=7x+6 has a unique rational solution.","1. Start with the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 3(5 - 2x) = 7x + 6 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n2. Distribute the 3 on the left side:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 15 - 6x = 7x + 6 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Rearrange the equation to bring all terms involving \\(x\\) to one side:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> -6x - 7x = 6 - 15 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Combine like terms:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> -13x = -9 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n5. Solve for \\(x\\) by dividing both sides by \\(-13\\):\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{-9}{-13} \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n6. Simplify the expression:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{9}{13} \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nThus, the unique rational solution is \\( x = \\frac{9}{13} 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