longmult 324.45*98.4

{ "query": { "display": "long multiplication $$324.45\\cdot\\:98.4$$", "symbolab_question": "PRE_ALGEBRA#longmult 324.45\\cdot 98.4" }, "solution": { "level": "PERFORMED", "subject": "Pre Algebra", "topic": "Decimals", "subTopic": "Long Multiplication", "default": "31925.88", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$324.45\\cdot\\:98.4=31925.88$$", "input": "324.45\\cdot\\:98.4", "steps": [ { "type": "step", "primary": "Multiply without the decimal points, then put the decimal point in the answer" }, { "type": "interim", "title": "$$32445\\cdot\\:984=31925880$$", "input": "32445\\cdot\\:984", "steps": [ { "type": "step", "primary": "Line up the numbers", "result": "\\begin{matrix}\\:\\:&3&2&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&9&8&4\\end{matrix}" }, { "type": "step", "primary": "Multiply the top number by the bottom number one digit at a time starting with the ones digit (from right to left)" }, { "type": "step", "primary": "Multiply the top number by the bolded digit of the bottom number", "result": "\\begin{matrix}\\:\\:&\\bold{3}&\\bold{2}&\\bold{4}&\\bold{4}&\\bold{5}\\\\\\times\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}" }, { "type": "step", "primary": "Mutliply the bold numbers: $$5\\cdot\\:4=20$$", "secondary": [ "Carry $$2$$ to the column on the left and write $$0$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&2&\\:\\:\\\\\\:\\:&3&2&4&4&\\bold{5}\\\\\\times\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&0\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$2+4\\cdot\\:4=18$$", "secondary": [ "Carry $$1$$ to the column on the left and write $$8$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&1&\\bold{2}&\\:\\:\\\\\\:\\:&3&2&4&\\bold{4}&5\\\\\\times\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$1+4\\cdot\\:4=17$$", "secondary": [ "Carry $$1$$ to the column on the left and write $$7$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&1&\\bold{1}&2&\\:\\:\\\\\\:\\:&3&2&\\bold{4}&4&5\\\\\\times\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&7&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$1+2\\cdot\\:4=9$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\bold{1}&1&2&\\:\\:\\\\\\:\\:&3&\\bold{2}&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&9&7&8&0\\end{matrix}}" }, { "type": "step", "primary": "Mutliply the bold numbers: $$3\\cdot\\:4=12$$", "secondary": [ "Carry $$1$$ to the column on the left and write $$2$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&1&\\:\\:&1&1&2&\\:\\:\\\\\\:\\:&\\:\\:&\\bold{3}&2&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&\\:\\:&9&8&\\bold{4}\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&2&9&7&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the carried digit, $$1$$, to the result", "result": "\\frac{\\begin{matrix}\\:\\:&1&\\:\\:&1&1&2&\\:\\:\\\\\\:\\:&\\:\\:&3&2&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&\\:\\:&9&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\end{matrix}}" }, { "type": "step", "primary": "Multiply the top number by the bolded digit of the bottom number", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\bold{3}&\\bold{2}&\\bold{4}&\\bold{4}&\\bold{5}\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\end{matrix}}" }, { "type": "step", "primary": "Mutliply the bold numbers: $$5\\cdot\\:8=40$$", "secondary": [ "Carry $$4$$ to the column on the left and write $$0$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&4&\\:\\:\\\\\\:\\:&\\:\\:&3&2&4&4&\\bold{5}\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$4+4\\cdot\\:8=36$$", "secondary": [ "Carry $$3$$ to the column on the left and write $$6$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&3&\\bold{4}&\\:\\:\\\\\\:\\:&\\:\\:&3&2&4&\\bold{4}&5\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\\\:\\:&\\:\\:&\\:\\:&\\:\\:&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$3+4\\cdot\\:8=35$$", "secondary": [ "Carry $$3$$ to the column on the left and write $$5$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&3&\\bold{3}&4&\\:\\:\\\\\\:\\:&\\:\\:&3&2&\\bold{4}&4&5\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\\\:\\:&\\:\\:&\\:\\:&5&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$3+2\\cdot\\:8=19$$", "secondary": [ "Carry $$1$$ to the column on the left and write $$9$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&1&\\bold{3}&3&4&\\:\\:\\\\\\:\\:&\\:\\:&3&\\bold{2}&4&4&5\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\\\:\\:&\\:\\:&9&5&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$1+3\\cdot\\:8=25$$", "secondary": [ "Carry $$2$$ to the column on the left and write $$5$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&2&\\bold{1}&3&3&4&\\:\\:\\\\\\:\\:&\\:\\:&\\bold{3}&2&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&\\:\\:&9&\\bold{8}&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\\\:\\:&5&9&5&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried digit, $$2$$, to the result", "result": "\\frac{\\begin{matrix}\\:\\:&2&1&3&3&4&\\:\\:\\\\\\:\\:&\\:\\:&3&2&4&4&5\\\\\\times\\:&\\:\\:&\\:\\:&\\:\\:&9&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\2&5&9&5&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Multiply the top number by the bolded digit of the bottom number", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\bold{3}&\\bold{2}&\\bold{4}&\\bold{4}&\\bold{5}\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&1&2&9&7&8&0\\\\2&5&9&5&6&0&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Mutliply the bold numbers: $$5\\cdot\\:9=45$$", "secondary": [ "Carry $$4$$ to the column on the left and write $$5$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&4&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&3&2&4&4&\\bold{5}\\\\\\:\\:&\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$4+4\\cdot\\:9=40$$", "secondary": [ "Carry $$4$$ to the column on the left and write $$0$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&4&\\bold{4}&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&3&2&4&\\bold{4}&5\\\\\\:\\:&\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&\\:\\:&0&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$4+4\\cdot\\:9=40$$", "secondary": [ "Carry $$4$$ to the column on the left and write $$0$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&4&\\bold{4}&4&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&3&2&\\bold{4}&4&5\\\\\\:\\:&\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&0&0&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$4+2\\cdot\\:9=22$$", "secondary": [ "Carry $$2$$ to the column on the left and write $$2$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&2&\\bold{4}&4&4&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&3&\\bold{2}&4&4&5\\\\\\:\\:&\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\\\:\\:&\\:\\:&2&0&0&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried number to the multiplication: $$2+3\\cdot\\:9=29$$", "secondary": [ "Carry $$2$$ to the column on the left and write $$9$$ in the result line" ], "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&2&\\bold{2}&4&4&4&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&\\bold{3}&2&4&4&5\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{9}&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\\\:\\:&9&2&0&0&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the carried digit, $$2$$, to the result", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&2&2&4&4&4&\\:\\:\\\\\\:\\:&\\:\\:&\\:\\:&3&2&4&4&5\\\\\\:\\:&\\times\\:&\\:\\:&\\:\\:&\\:\\:&9&8&4\\end{matrix}}{\\begin{matrix}\\:\\:&\\:\\:&1&2&9&7&8&0\\\\\\:\\:&2&5&9&5&6&0&\\:\\:\\\\2&9&2&0&0&5&\\:\\:&\\:\\:\\end{matrix}}" }, { "type": "step", "primary": "Add the rows to get the answer. For simplicity, fill in trailing zeros ", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&3&2&4&4&5\\\\\\:\\:&\\:\\:&\\times\\:&\\:\\:&\\:\\:&9&8&4\\end{matrix}}{\\begin{matrix}0&0&1&2&9&7&8&0\\\\0&2&5&9&5&6&0&0\\\\2&9&2&0&0&5&0&0\\end{matrix}}" }, { "type": "interim", "title": "$$129780+2595600+29200500=31925880$$", "input": "\\begin{matrix}\\:\\:&0&0&1&2&9&7&8&0\\\\\\:\\:&0&2&5&9&5&6&0&0\\\\+&2&9&2&0&0&5&0&0\\end{matrix}", "steps": [ { "type": "step", "primary": "Add the digits of the bolded column: $$0+0+0=0$$", "result": "\\frac{\\begin{matrix}\\:\\:&0&0&1&2&9&7&8&\\bold{0}\\\\\\:\\:&0&2&5&9&5&6&0&\\bold{0}\\\\+&2&9&2&0&0&5&0&\\bold{0}\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{0}\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$8+0+0=8$$", "result": "\\frac{\\begin{matrix}\\:\\:&0&0&1&2&9&7&\\bold{8}&0\\\\\\:\\:&0&2&5&9&5&6&\\bold{0}&0\\\\+&2&9&2&0&0&5&\\bold{0}&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{8}&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$7+6+5=18$$", "result": "\\frac{\\begin{matrix}\\:\\:&0&0&1&2&9&\\bold{7}&8&0\\\\\\:\\:&0&2&5&9&5&\\bold{6}&0&0\\\\+&2&9&2&0&0&\\bold{5}&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{\\:\\:}&8&0\\end{matrix}}" }, { "type": "step", "primary": "Carry $$1$$ to the column on the left and write $$8$$ in the bolded column", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&1&\\bold{\\:\\:}&\\:\\:&\\:\\:\\\\\\:\\:&0&0&1&2&9&\\bold{7}&8&0\\\\\\:\\:&0&2&5&9&5&\\bold{6}&0&0\\\\+&2&9&2&0&0&\\bold{5}&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{8}&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$1+9+5+0=15$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{1}&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&0&1&2&\\bold{9}&7&8&0\\\\\\:\\:&0&2&5&9&\\bold{5}&6&0&0\\\\+&2&9&2&0&\\bold{0}&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{\\:\\:}&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Carry $$1$$ to the column on the left and write $$5$$ in the bolded column", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&1&\\bold{1}&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&0&1&2&\\bold{9}&7&8&0\\\\\\:\\:&0&2&5&9&\\bold{5}&6&0&0\\\\+&2&9&2&0&\\bold{0}&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{5}&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$1+2+9+0=12$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\:\\:&\\bold{1}&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&0&1&\\bold{2}&9&7&8&0\\\\\\:\\:&0&2&5&\\bold{9}&5&6&0&0\\\\+&2&9&2&\\bold{0}&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\bold{\\:\\:}&5&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Carry $$1$$ to the column on the left and write $$2$$ in the bolded column", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&1&\\bold{1}&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&0&1&\\bold{2}&9&7&8&0\\\\\\:\\:&0&2&5&\\bold{9}&5&6&0&0\\\\+&2&9&2&\\bold{0}&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\:\\:&\\bold{2}&5&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$1+1+5+2=9$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\:\\:&\\bold{1}&1&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&0&\\bold{1}&2&9&7&8&0\\\\\\:\\:&0&2&\\bold{5}&9&5&6&0&0\\\\+&2&9&\\bold{2}&0&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\:\\:&\\bold{9}&2&5&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$0+2+9=11$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\:\\:&\\bold{\\:\\:}&1&1&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&\\bold{0}&1&2&9&7&8&0\\\\\\:\\:&0&\\bold{2}&5&9&5&6&0&0\\\\+&2&\\bold{9}&2&0&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\bold{\\:\\:}&9&2&5&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Carry $$1$$ to the column on the left and write $$1$$ in the bolded column", "result": "\\frac{\\begin{matrix}\\:\\:&1&\\bold{\\:\\:}&1&1&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&0&\\bold{0}&1&2&9&7&8&0\\\\\\:\\:&0&\\bold{2}&5&9&5&6&0&0\\\\+&2&\\bold{9}&2&0&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\:\\:&\\bold{1}&9&2&5&8&8&0\\end{matrix}}" }, { "type": "step", "primary": "Add the digits of the bolded column: $$1+0+0+2=3$$", "result": "\\frac{\\begin{matrix}\\:\\:&\\bold{1}&\\:\\:&1&1&1&\\:\\:&\\:\\:&\\:\\:\\\\\\:\\:&\\bold{0}&0&1&2&9&7&8&0\\\\\\:\\:&\\bold{0}&2&5&9&5&6&0&0\\\\+&\\bold{2}&9&2&0&0&5&0&0\\end{matrix}}{\\begin{matrix}\\phantom{+}&\\bold{3}&1&9&2&5&8&8&0\\end{matrix}}" }, { "type": "step", "result": "=31925880" } ], "meta": { "solvingClass": "StepsLongAddition", "interimType": "StepsLongAddition" } }, { "type": "step", "result": "=31925880" } ], "meta": { "solvingClass": "StepsLongMultiplication", "interimType": "StepsLongMultiplication" } }, { "type": "step", "primary": "$$324.45$$ has $$2$$ decimal places", "secondary": [ "$$98.4$$ has $$1$$ decimal place", "Therefore, the answer has $$3$$ decimal places" ], "result": "=31925.880" }, { "type": "step", "primary": "Refine", "result": "=31925.88" } ], "meta": { "solvingClass": "StepsLongMultiplication" } }, "meta": { "showVerify": true } }