Video: Computing Unit Rates Associated with Ratios of Fractions

A box-shaped piece of steel of dimensions 4 cm × 2/5 cm × 1/2 cm weighs 6 2/5 g. What is its density (i.e., the mass of 1 cm³)?

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Video Transcript

A box-shaped piece of steel of dimensions four centimeters by two-fifths of a centimeter by a half a centimeter weighs six and two-fifths grams. What is its density, i.e., the mass of one cubic centimeter?

The density of any object can be calculated by dividing its mass by its volume. You might have seen the link between density, mass, and volume written as in the triangle shown. In this question, our mass is measured in grams. And as the dimensions are in centimeters, the volume will be in cubic centimeters or centimeters cubed. This means that our units for density will be grams per centimeter cubed.

As our steel is box shaped, it is in the shape of a rectangular prism, sometimes known as a cuboid. To calculate the volume of this, we multiply the length by the width by the height. We need to multiply four by two-fifths by one-half. We can simplify this by cross-canceling. We can divide the two in the numerator and the two on the denominator by two. This leaves us with a volume of four-fifths. Our units will be cubic centimeters or centimeters cubed.

As the mass of the steel is six and two-fifths grams, we can calculate the density by dividing this by four-fifths. A mixed number can be converted into an improper or top heavy fraction by multiplying the whole number by the denominator and then adding the numerator. Six multiplied by five is 30, and adding two is 32.

We need to divide thirty-two fifths by four-fifths. Dividing by a fraction is the same as multiplying by the reciprocal of this fraction. We could find the reciprocal four-fifths and then multiply the two fractions. However, as the denominators are the same, it will be easier to just cancel these. We are left with 32 divided by four, which is equal to eight.

The density of the steel is eight grams per centimeter cubed or eight grams per cubic centimeter.

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