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Matthew was collecting small change for charity.
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He collected cents, nickels, and dimes.
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He collected one and a half times as many cents as nickels and 38 more dimes than nickels.
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There are 388 coins in his collection box.
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How much money did he collect altogether?
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Letβs start with the three types of coins.
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Matthew collected cents, nickels, and dimes.
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Weβll let π₯ be the number of nickels that he collected.
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Weβre told in the question that Matthew collected one and a half times as many cents as nickels.
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This is equal to 1.5π₯ as one and a half is the same as 1.5.
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Matthew also collected 38 more dimes than nickels.
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As there were π₯ nickels, thereβre π₯ plus 38 dimes.
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Adding these three expressions gives us 1.5π₯ plus π₯ plus π₯ plus 38.
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We were told that Matthew had 388 coins in his collection box.
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Therefore, this expression is equal to 388.
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Collecting the like terms on the left-hand side gives us 3.5π₯ as 1.5π₯ plus π₯ plus π₯ is 3.5π₯.
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We, therefore, have 3.5π₯ plus 38 equals 388.
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Subtracting 38 from both sides of this equation gives us 3.5π₯ is equal to 350 as 388 minus 38 equals 350.
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Dividing both sides of this equation by 3.5 gives us a value of π₯ equal to 100.
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As π₯ is equal to 100, we can now work out how many cents, nickels, and dimes Matthew had.
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As the number of nickels was equal to π₯, Matthew has 100 nickels.
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1.5 multiplied by 100 is equal to 150.
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Therefore, Matthew has 150 cents.
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100 plus 38 is equal to 138.
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Therefore, Matthew has 138 dimes.
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We can check at this stage that these three numbers add up to 388.
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150 plus 100 is 250.
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Adding 138 to this gives us 388.
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We were asked to calculate how much money Matthew had altogether.
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In order to do this, we need to multiply 150 by 0.01.
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We need to multiply 100 by 0.05 and we need to multiply 138 by 0.1 or 0.10.
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This is because there are five cents in a nickel and there are 10 cents in a dime.
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150 multiplied by 0.1 is equal to 1.5.
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Therefore, Matthew has one dollar and 50 cents worth of cents.
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100 multiplied by 0.05 is equal to five.
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Therefore, Matthew has five dollars worth of nickels.
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138 multiplied by 0.1 is equal to 13.8.
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Therefore, Matthew has 13 dollars and 80 cents of dimes.
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In order to calculate the total amount of money that Matthew collected, we need to add one dollar and 50 cents to five dollars to 13 dollars and 80 cents.
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This is equal 20 dollars and 30 cents.
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Altogether, Matthew has 20 dollars and 30 cents in his collection box.
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This is made up of cents, nickels, and dimes.